Eikonal Blog


Denialism of science

  • Denialism (WikiPedia) – http://en.wikipedia.org/wiki/Denialism
    • In human behavior, denialism is exhibited by individuals choosing to deny reality as a way to avoid dealing with an uncomfortable truth. … “[It] is the refusal to accept an empirically verifiable reality. It is an essentially irrational action that withholds validation of a historical experience or event”. … group denialism [is defined] as “when an entire segment of society, often struggling with the trauma of change, turns away from reality in favor of a more comfortable lie.”
    • In science, denialism has been defined as the rejection of basic concepts that are undisputed and well-supported parts of the scientific consensus on a topic in favor of ideas that are both radical and controversial. … It has been proposed that the various forms of denialism have the common feature of the rejection of overwhelming evidence and the generation of a controversy through attempts to deny that a consensus exists. … A common example is Young Earth creationism and its dispute with the evolutionary theory.


  • “How To Convince Conservative Christians That Global Warming Is Real” by Chris Mooney (Mother Jones; 2014.05.02) – http://www.motherjones.com/environment/2014/05/inquiring-minds-katharine-hayhoe-faith-climate
    • Millions of Americans are evangelical Christians. Climate scientist Katharine Hayhoe is persuading them that our planet is in peril.
    • “Years of Living Dangerously Premiere Full Episode” – https://www.youtube.com/watch?v=brvhCnYvxQQ
  • “Most Americans doubt Big Bang, not too sure about evolution, climate change – survey” By Rik Myslewski (The Register; 2014.04.21) – http://www.theregister.co.uk/2014/04/21/most_americans_doubt_big_bang_not_too_sure_about_evolution_climate_change_survey/
    • Science no match for religion, politics, business interests


  • “AP-GfK Poll: Big Bang a big question for most Americans” (AP-Gfk; 2014.04.21) – http://ap-gfkpoll.com/featured/findings-from-our-latest-poll-2
    • Few Americans question that smoking causes cancer. But they express bigger doubts as concepts that scientists consider to be truths get further from our own experiences and the present time … Americans have more skepticism than confidence in global warming, the age of the Earth and evolution and have the most trouble believing a Big Bang created the universe 13.8 billion years ago….
    • Just 4 percent doubt that smoking causes cancer, 6 percent question whether mental illness is a medical condition that affects the brain and 8 percent are skeptical there’s a genetic code inside our cells. More – 15 percent – have doubts about the safety and efficacy of childhood vaccines …
    • About 4 in 10 say they are not too confident or outright disbelieve that the earth is warming, mostly a result of man-made heat-trapping gases, that the Earth is 4.5 billion years old or that life on Earth evolved through a process of natural selection, though most were at least somewhat confident in each of those concepts. But a narrow majority – 51 percent – questions the Big Bang theory …
    • “Science ignorance is pervasive in our society, and these attitudes are reinforced when some of our leaders are openly antagonistic to established facts,”…
    • The poll highlights “the iron triangle of science, religion and politics,” … And scientists know they’ve got the shakiest leg in the triangle….
    • To the public “most often values and beliefs trump science” when they conflict, … … Political values were closely tied to views on science in the poll, with Democrats more apt than Republicans to express confidence in evolution, the Big Bang, the age of the Earth and climate change….
    • Religious values are similarly important… Confidence in evolution, the Big Bang, the age of the Earth and climate change decline sharply as faith in a supreme being rises, according to the poll. Likewise, those who regularly attend religious services or are evangelical Christians express much greater doubts about scientific concepts they may see as contradictory to their faith … “When you are putting up facts against faith, facts can’t argue against faith,” … “It makes sense now that science would have made no headway because faith is untestable.” …
    • Beyond religious belief, views on science may be tied to what we see with our own eyes. The closer an issue is to our bodies and the less complicated, the easier it is for people to believe, …
    • Marsha Brooks, a 59-year-old nanny who lives in Washington, D.C., said she’s certain smoking causes cancer because she saw her mother, aunts and uncles, all smokers, die of cancer … But when it comes to the universe beginning with a Big Bang or the Earth being about 4.5 billion years old, she has doubts. …
    • Jorge Delarosa, a 39-year-old architect from Bridgewater, N.J., pointed to a warm 2012 without a winter and said, “I feel the change. There must be a reason.” But when it came to Earth’s beginnings 4.5 billion years ago, he has doubts simply because “I wasn’t there.”…
    • Experience and faith aren’t the only things affecting people’s views on science. … “the force of concerted campaigns to discredit scientific fact” as a more striking factor, citing significant interest groups – political, business and religious – campaigning against scientific truths on vaccines, climate change and evolution….
    • … sometimes science wins out even against well-financed and loud opposition, as with smoking. Widespread belief that smoking causes cancer “has come about because of very public, very focused public health campaigns,” … [also, what is very encouraging is] the public’s acceptance that mental illness is a brain disease, something few believed 25 years ago, before just such a campaign.
  • “Why climate deniers are winning: The twisted psychology that overwhelms scientific consensus” by Paul Rosenberg (The Salon; 2014.04.19) – http://www.salon.com/2014/04/19/why_climate_deniers_are_winning_the_twisted_psychology_that_overwhelms_scientific_consensus/
    • There’s a reason why overwhelming evidence hasn’t spurred public action against global warming
    • “The reason ‘consensus’ has not appeared to work in society at large to date isn’t because it’s ineffective – it’s because there is a well-funded counter-movement out there that takes every opportunity to mislead the public into thinking that there isn’t a consensus,”
  • “How politics makes us stupid” by Ezra Klein (Vox; 2014.04.06) – http://www.vox.com/2014/4/6/5556462/brain-dead-how-politics-makes-us-stupid

Older articles

  • “How Do You Get People to Give a Damn About Climate Change?” by Chris Mooney (Mother Jones; 2013.10.18) – http://www.motherjones.com/environment/2013/10/inquiring-minds-kahan-lewandowsky-communicate-climate
    • Experts have come a long way in figuring out which messages can successfully open minds and move public opinion. There’s just one problem: They disagree about whether the message everyone’s using actually works.
  • “Scientific uncertainty and climate change: Part I. Uncertainty and unabated emissions” by Stephan Lewandowsky, James S. Risbey, Michael Smithson, Ben R. Newell, John Hunter (Springer) – http://link.springer.com/article/10.1007/s10584-014-1082-7
    • Uncertainty forms an integral part of climate science, and it is often used to argue against mitigative action. This article presents an analysis of uncertainty in climate sensitivity that is robust to a range of assumptions. We show that increasing uncertainty is necessarily associated with greater expected damages from warming, provided the function relating warming to damages is convex. This constraint is unaffected by subjective or cultural risk-perception factors, it is unlikely to be overcome by the discount rate, and it is independent of the presumed magnitude of climate sensitivity. The analysis also extends to “second-order” uncertainty; that is, situations in which experts disagree. Greater disagreement among experts increases the likelihood that the risk of exceeding a global temperature threshold is greater. Likewise, increasing uncertainty requires increasingly greater protective measures against sea level rise. This constraint derives directly from the statistical properties of extreme values. We conclude that any appeal to uncertainty compels a stronger, rather than weaker, concern about unabated warming than in the absence of uncertainty.
  • “Scientific uncertainty and climate change: Part II. Uncertainty and mitigation” by Stephan Lewandowsky, James S. Risbey, Michael Smithson, Ben R. Newell (Springer) – http://link.springer.com/article/10.1007/s10584-014-1083-6
    • In public debate surrounding climate change, scientific uncertainty is often cited in connection with arguments against mitigative action. This article examines the role of uncertainty about future climate change in determining the likely success or failure of mitigative action. We show by Monte Carlo simulation that greater uncertainty translates into a greater likelihood that mitigation efforts will fail to limit global warming to a target (e.g., 2 °C). The effect of uncertainty can be reduced by limiting greenhouse gas emissions. Taken together with the fact that greater uncertainty also increases the potential damages arising from unabated emissions (Lewandowsky et al. 2014), any appeal to uncertainty implies a stronger, rather than weaker, need to cut greenhouse gas emissions than in the absence of uncertainty.
  • “The pivotal role of perceived scientific consensus in acceptance of science” by Stephan Lewandowsky, Gilles E. Gignac, Samuel Vaughan (Nature Climate Change 3, 399-404 (2013); doi:10.1038/nclimate1720; 2012.10.28) – http://www.nature.com/nclimate/journal/v3/n4/full/nclimate1720.html
    • Although most experts agree that CO2 emissions are causing anthropogenic global warming (AGW), public concern has been declining. One reason for this decline is the ‘manufacture of doubt’ by political and vested interests, which often challenge the existence of the scientific consensus. The role of perceived consensus in shaping public opinion is therefore of considerable interest: in particular, it is unknown whether consensus determines people’s beliefs causally. It is also unclear whether perception of consensus can override people’s ‘worldviews’, which are known to foster rejection of AGW. Study 1 shows that acceptance of several scientific propositions-from HIV/AIDS to AGW-is captured by a common factor that is correlated with another factor that captures perceived scientific consensus. Study 2 reveals a causal role of perceived consensus by showing that acceptance of AGW increases when consensus is highlighted. Consensus information also neutralizes the effect of worldview.



Reality of wave function in quantum mechanics

Filed under: physics, Quantum mechanics — Tags: , , — sandokan65 @ 15:39



  • “When the multiverse and many-worlds collide” by Justin Mullins (The New Scientist; 2011.06.01) – http://www.newscientist.com/article/mg21028154.200-when-the-multiverse-and-manyworlds-collide.html
  • “Are Many Worlds and the Multiverse the Same Idea?” by Sean Carroll (Cosmic Variance blog at Discover Magazine; ) – http://blogs.discovermagazine.com/cosmicvariance/2011/05/26/are-many-worlds-and-the-multiverse-the-same-idea/
  • “Physical Theories, Eternal Inflation, and Quantum Universe” by Yasunori Nomura (arXiv.org > hep-th > arXiv:1104.2324v2 [hep-th])- http://arxiv.org/abs/1104.2324
      Abstract: We present a framework in which well-defined predictions are obtained in an eternally inflating multiverse, based on the principles of quantum mechanics. We show that the entire multiverse is described purely from the viewpoint of a single “observer,” who describes the world as a quantum state defined on his/her past light cones bounded by the (stretched) apparent horizons. We find that quantum mechanics plays an essential role in regulating infinities. The framework is “gauge invariant,” i.e. predictions do not depend on how spacetime is parametrized, as it should be in a theory of quantum gravity. Our framework provides a fully unified treatment of quantum measurement processes and the multiverse. We conclude that the eternally inflating multiverse and many worlds in quantum mechanics are the same. Other important implications include: global spacetime can be viewed as a derived concept; the multiverse is a transient phenomenon during the world relaxing into a supersymmetric Minkowski state. We also present a theory of “initial conditions” for the multiverse. By extrapolating our framework to the extreme, we arrive at a picture that the entire multiverse is a fluctuation in the stationary, fractal “mega-multiverse,” in which an infinite sequence of multiverse productions occurs. The framework discussed here does not suffer from problems/paradoxes plaguing other measures proposed earlier, such as the youngness paradox, the Boltzmann brain problem, and a peculiar “end” of time.
  • “The Multiverse Interpretation of Quantum Mechanics” by Raphael Bousso and Leonard Susskind (arXiv.org > hep-th > arXiv:1105.3796v1 [hep-th]) – http://arxiv.org/abs/1105.3796
      Abstract: We argue that the many-worlds of quantum mechanics and the many worlds of the multiverse are the same thing, and that the multiverse is necessary to give exact operational meaning to probabilistic predictions from quantum mechanics.
      Decoherence – the modern version of wave-function collapse – is subjective in that it depends on the choice of a set of unmonitored degrees of freedom, the “environment”. In fact decoherence is absent in the complete description of any region larger than the future light-cone of a measurement event. However, if one restricts to the causal diamond – the largest region that can be causally probed – then the boundary of the diamond acts as a one-way membrane and thus provides a preferred choice of environment. We argue that the global multiverse is a representation of the many-worlds (all possible decoherent causal diamond histories) in a single geometry.
      We propose that it must be possible in principle to verify quantum-mechanical predictions exactly. This requires not only the existence of exact observables but two additional postulates: a single observer within the universe can access infinitely many identical experiments; and the outcome of each experiment must be completely definite. In causal diamonds with finite surface area, holographic entropy bounds imply that no exact observables exist, and both postulates fail: experiments cannot be repeated infinitely many times; and decoherence is not completely irreversible, so outcomes are not definite. We argue that our postulates can be satisfied in “hats” (supersymmetric multiverse regions with vanishing cosmological constant). We propose a complementarity principle that relates the approximate observables associated with finite causal diamonds to exact observables in the hat.


Astronomy & Astrophysics

Filed under: astronomy, cosmology, physics — sandokan65 @ 10:16


  • “Kepler May Uncover Numerous Ring Worlds” 9SlashDot; 2011.05.08) – http://science.slashdot.org/story/11/05/08/2230208/Kepler-May-Uncover-Numerous-Ring-Worlds
      “According to a new publication, NASA’s Kepler exoplanet-hunting space telescope may soon start discovering Saturn-like ringed alien worlds. So far, none have been positively identified, as Kepler has only detected exoplanets orbiting close to their parent stars; if these exoplanets have rings, they are most likely to have rings facing edge-on to their orbits, making them nearly impossible to detect. As more distant-orbiting exoplanets are detected, there’s more likelihood ringed worlds will be tilted, allowing Kepler to see them.”

Related here: Astronomic tables and calculators – https://eikonal.wordpress.com/2010/01/28/astronomic-tables-and-calculators/ | Solar planetary system – https://eikonal.wordpress.com/2011/02/17/solar-planetary-system/ | Inside black holes – https://eikonal.wordpress.com/2011/04/12/inside-black-holes/ | Physics Sites – https://eikonal.wordpress.com/2010/02/12/physics-sites/


PT-Quantum Mechanics

Filed under: physics, Quantum mechanics — Tags: , , , , — sandokan65 @ 14:39
  • Carl Bender’s speaches at PIRSA (Perimeter Institute Recorded Seminar Archive) – http://pirsa.org/index.php?p=speaker&name=Carl_Bender
  • Carl Bender’s talk on turbulence, iterated maps, chaos and classical mechanics in classical domain. Includes several interesting experimental demonstrations. – http://artsci.wustl.edu/~spenteco/newVideo/DLP_bender_083109.html
  • Carl Bender’s papers at arXiv – http://arxiv.org/find/quant-ph/1/au:+Bender_C/0/1/0/all/0/1
    • “Response to Shalaby’s Comment on “Families of Particles with Different Masses in PT-Symmetric Quantum Field Theory”” by Carl M. Bender, S. P. Klevansky (arXiv:1103.0338; 2011.03.02) – http://arxiv.org/abs/1103.0338
        Abstract: In a recent Comment [arXiv: 1101.3980] Shalaby criticised our paper “Families of Particles with Different Masses in PT-Symmetric Quantum Field Theory” [arXiv:1002.3253]. On examining his arguments, we find that there are serious flaws at almost every stage of his Comment. In view of space and time considerations, we point out the major flaws that render his arguments invalid. Essentially Shalaby is attempting to obtain our results from a variational principle and to find a physical interpretation of his calculation. The variational procedure that he uses is inapplicable, and his description of the physics is wrong. We thus refute his criticism on all levels.
    • “PT-symmetric quantum state discrimination” by Carl M. Bender, Dorje C. Brody, Joao Caldeira, Bernard K. Meister (arXiv:1011.1871; 2010.11.08) – http://arxiv.org/abs/1011.1871
        Abstract: Suppose that a system is known to be in one of two quantum states, |\psi_1 > or |\psi_2 >. If these states are not orthogonal, then in conventional quantum mechanics it is impossible with one measurement to determine with certainty which state the system is in. However, because a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states, it is always possible to choose this inner product so that the two states |\psi_1 > and |\psi_2 > are orthogonal. Thus, quantum state discrimination can, in principle, be achieved with a single measurement.
    • “Tunneling in classical mechanics” by Carl M. Bender, Daniel W. Hook (arXiv:1011.0121; 2010.09.16) – http://arxiv.org/abs/1011.0121
        Abstract: A classical particle that is initially in a classically allowed region of a potential is not confined to this region for all time if its energy is complex. Rather, the particle may travel through complex coordinate space and visit other classically allowed regions. Thus, a complex-energy classical particle can exhibit tunneling-like behavior. This tunneling behavior persists as the imaginary part of the energy tends to zero. Hence one may compare complex classical tunneling times with quantum tunneling probabilities. An accurate numerical study of quantum and classical tunneling demonstrates that as the energy increases, the probabilities associated with complex classical tunneling approach the corresponding quantum probabilities.
      • The Hamiltonian H=p^2+V^{(4)}(x) with the potential V^{(4)}(x) = \frac72 x (x-1) \left(x+\frac{191}{100}\right)\left(x-\frac{49}{20}\right) has following lowest energy levels:
        • E_0 = - 18.018,2,
        • E_1 = - 7.187,9,
        • E_2 = - 6.859,5,
        • E_3 = + 1.680,6,
        • E_4 = + 2.884,5.
        • E_5 = + 8.331,2 lies just above the barrier separating two potential wells.
      • The sextic potential V^{(6)}(x) = x^6 - 2 x^5 -4 x^4 + 11 x^3 - \frac{11}4 x^2 -13 x has following lowest energy levels:
        • E_0 = - 23/2,
        • E_1 = - 9.969,0,
        • E_2 = - 3.981,9,
        • E_3 = + 1.809,5.
      • The geometric charactericts of the quartic potential are:
        • the left well minimum is at (x=-1.2499, V=-24.0384);
        • the right well minimum is at (x=+1.9165, V=-12.5501);
        • the barrier is at (x=+0.4884, V=+4.1144)
      • The geometric charactericts of the sextic potential are:
        • the left well minimum is at (x=-1.7083, V=-20.7710);
        • the right well minimum is at (x=+1.8215, V=-13.9373);
        • the barrier is at (x=-0.5184, V=+4.2731)
    • “Extending PT symmetry from Heisenberg algebra to E2 algebra” by Carl M. Bender, R. J. Kalveks (arXiv:1009.3236 2010.09.16; Int.J.Theor.Phys.50:955-962,2011) – http://arxiv.org/abs/1009.3236
        Abstract: The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.
    • “Quantum counterpart of spontaneously broken classical PT symmetry” by Carl M. Bender, Hugh F. Jones (arXiv:1008.0782 2010.08.04; J.Phys.A44:015301,2011) – http://arxiv.org/abs/1008.0782
        Abstract: The classical trajectories of a particle governed by the PT-symmetric Hamiltonian H=p^2+x^2(ix)^\epsilon (\epsilon\geq0) have been studied in depth. It is known that almost all trajectories that begin at a classical turning point oscillate periodically between this turning point and the corresponding PT-symmetric turning point. It is also known that there are regions in \epsilon for which the periods of these orbits vary rapidly as functions of \epsilon and that in these regions there are isolated values of \epsilon for which the classical trajectories exhibit spontaneously broken PT symmetry. The current paper examines the corresponding quantum-mechanical systems. The eigenvalues of these quantum systems exhibit characteristic behaviors that are correlated with those of the associated classical system.
    • “Almost zero-dimensional PT-symmetric quantum field theories” by Carl M. Bender (arXiv:1003.3881; 2010.03.19) – http://arxiv.org/abs/1003.3881
        Abstract: In 1992 Bender, Boettcher, and Lipatov proposed in two papers a new and unusual nonperturbative calculational tool in quantum field theory. The objective was to expand the Green’s functions of the quantum field theory as Taylor series in powers of the space-time dimension D. In particular, the vacuum energy for a massless \phi^{2N} (N=1,2,3,…) quantum field theory was studied. The first two Taylor coefficients in this dimensional expansion were calculated {\it exactly} and a set of graphical rules were devised that could be used to calculate approximately the higher coefficients in the series. This approach is mathematically valid and gives accurate results, but it has not been actively pursued and investigated. Subsequently, in 1998 Bender and Boettcher discovered that PT-symmetric quantum-mechanical Hamiltonians of the form H=p^2+x^2(ix)^\epsilon, where \epsilon\geq0, have real spectra. These new kinds of complex non-Dirac-Hermitian Hamiltonians define physically acceptable quantum-mechanical theories. This result in quantum mechanics suggests that the corresponding non-Dirac-Hermitian D-dimensional \phi^2(i\phi)^\epsilon quantum field theories might also have real spectra. To examine this hypothesis, we return to the technique devised in 1992 and in this paper we calculate the first two coefficients in the dimensional expansion of the ground-state energy of this complex non-Dirac-Hermitian quantum field theory. We show that to first order in this dimensional approximation the ground-state energy is indeed real for \epsilon\geq0.
    • “Families of particles with different masses in PT-symmetric quantum field theory” by C. M. Bender, S. P. Klevansky (arXiv:1002.3253; 2010.07.05) – http://arxiv.org/abs/1002.3253
        Abstract: An elementary field-theoretic mechanism is proposed that allows one Lagrangian to describe a family of particles having different masses but otherwise similar physical properties. The mechanism relies on the observation that the Dyson-Schwinger equations derived from a Lagrangian can have many different but equally valid solutions. Nonunique solutions to the Dyson-Schwinger equations arise when the functional integral for the Green’s functions of the quantum field theory converges in different pairs of Stokes’ wedges in complex field space, and the solutions are physically viable if the pairs of Stokes’ wedges are PT symmetric.
    • “Classical Particle in a Complex Elliptic Potential” by Carl M. Bender, Daniel W. Hook, Karta Singh Kooner (arXiv:1001.1548 2010.01.10; J.Phys.A43:165201,2010) – http://arxiv.org/abs/1001.1548
        Abstract: This paper reports a numerical study of complex classical trajectories of a particle in an elliptic potential. This study of doubly-periodic potentials is a natural sequel to earlier work on complex classical trajectories in trigonometric potentials. For elliptic potentials there is a two-dimensional array of identical cells in the complex plane, and each cell contains a pair of turning points. The particle can travel both horizontally and vertically as it visits these cells, and sometimes the particle is captured temporarily by a pair of turning points. If the particle’s energy lies in a conduction band, the particle drifts through the lattice of cells and is never captured by the same pair of turning points more than once. However, if the energy of the particle is not in a conduction band, the particle can return to previously visited cells.
    • “Complex Elliptic Pendulum” by Carl M. Bender, Daniel W. Hook, Karta Kooner (arXiv:1001.0131; 2009.12.31) – http://arxiv.org/abs/1001.0131
        Abstract: This paper briefly summarizes previous work on complex classical mechanics and its relation to quantum mechanics. It then introduces a previously unstudied area of research involving the complex particle trajectories associated with elliptic potentials.
    • “Probability Density in the Complex Plane” by Carl M. Bender, Daniel W. Hook, Peter N. Meisinger, Qing-hai Wang (arXiv:0912.4659; 2010.01.23) – http://arxiv.org/abs/0912.4659
        Abstract: The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density $\rho(z)$ in the complex plane. It is shown that there exist infinitely many complex contours C of infinite length on which \rho(z) dz is real and positive. Furthermore, the probability integral \int_C\rho(z) dz is finite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders.
    • “Complex Correspondence Principle” by Carl M. Bender, Daniel W. Hook, Peter N. Meisinger, Qing-hai Wang (arXiv:0912.2069 2009.12.10; Phys.Rev.Lett.104:061601,2010) – http://arxiv.org/abs/0912.2069
        Abstract: Quantum mechanics and classical mechanics are two very different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on extending both quantum mechanics and classical mechanics into the complex domain. This letter shows that these complex extensions continue to exhibit a correspondence, and that this correspondence becomes more pronounced in the complex domain. The association between complex quantum mechanics and complex classical mechanics is subtle and demonstrating this relationship prequires the use of asymptotics beyond all orders.
    • “PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues” by Carl M. Bender, Philip D. Mannheim (arXiv:0902.1365; 2009.02.09) – http://arxiv.org/abs/0902.1365
        Abstract: Despite its common use in quantum theory, the mathematical requirement of Dirac Hermiticity of a Hamiltonian is sufficient to guarantee the reality of energy eigenvalues but not necessary. By establishing three theorems, this paper gives physical conditions that are both necessary and sufficient. First, it is shown that if the secular equation is real, the Hamiltonian is necessarily PT symmetric. Second, if a linear operator C that obeys the two equations [C,H]=0 and C^2=1 is introduced, then the energy eigenvalues of a PT-symmetric Hamiltonian that is diagonalizable are real only if this C operator commutes with PT. Third, the energy eigenvalues of PT-symmetric Hamiltonians having a nondiagonalizable, Jordan-block form are real. These theorems hold for matrix Hamiltonians of any dimensionality.
    • “Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians” by Carl M. Bender, Dorje C. Brody (arXiv:0808.1823; 2008.08.13) – http://arxiv.org/abs/0808.1823
        Abstract: Consider the set of all Hamiltonians whose largest and smallest energy eigenvalues, E_max and E_min, differ by a fixed energy \omega. Given two quantum states, an initial state |\psi_I> and a final state |\psi_F>, there exist many Hamiltonians H belonging to this set under which |\psi_I> evolves in time into |\psi_F>. Which Hamiltonian transforms the initial state to the final state in the least possible time \tau? For Hermitian Hamiltonians, \tau has a nonzero lower bound. However, among complex non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, \tau can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of \tau can be made arbitrarily small because for PT-symmetric Hamiltonians the evolution path from the vector |\psi_I> to the vector |\psi_F>, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here resembles the effect in general relativity in which two space-time points can be made arbitrarily close if they are connected by a wormhole. This result may have applications in quantum computing.
    • “Quantum effects in classical systems having complex energy” by Carl M. Bender, Dorje C. Brody, Daniel W. Hook (arXiv:0804.4169 2008.04.25; J.Phys.A41:352003,2008) – http://arxiv.org/abs/0804.4169
        Abstract: On the basis of extensive numerical studies it is argued that there are strong analogies between the probabilistic behavior of quantum systems defined by Hermitian Hamiltonians and the deterministic behavior of classical mechanical systems extended into the complex domain. Three models are examined: the quartic double-well potential V(x)=x^4-5x^2, the cubic potential V(x)=frac{1}{2}x^2-gx^3, and the periodic potential V(x)=-\cos x. For the quartic potential a wave packet that is initially localized in one side of the double-well can tunnel to the other side. Complex solutions to the classical equations of motion exhibit a remarkably analogous behavior. Furthermore, classical solutions come in two varieties, which resemble the even-parity and odd-parity quantum-mechanical bound states. For the cubic potential, a quantum wave packet that is initially in the quadratic portion of the potential near the origin will tunnel through the barrier and give rise to a probability current that flows out to infinity. The complex solutions to the corresponding classical equations of motion exhibit strongly analogous behavior. For the periodic potential a quantum particle whose energy lies between -1 and 1 can tunnel repeatedly between adjacent classically allowed regions and thus execute a localized random walk as it hops from region to region. Furthermore, if the energy of the quantum particle lies in a conduction band, then the particle delocalizes and drifts freely through the periodic potential. A classical particle having complex energy executes a qualitatively analogous local random walk, and there exists a narrow energy band for which the classical particle becomes delocalized and moves freely through the potential.
    • “Comment on the Quantum Brachistochrone Problem” by C. M. Bender, D. C. Brody, H. F. Jones, B. K. Meister (arXiv:0804.3487; 2008.04.22) – http://arxiv.org/abs/0804.3487
        Abstract: In this brief comment we attempt to clarify the apparent discrepancy between the papers [1] and [2] on the quantum brachistochrone, namely whether it is possible to use a judicious mixture of Hermitian and non-Hermitian quantum mechanics to evade the standard lower limit on the time taken for evolution by a Hermitian Hamiltonian with given energy dispersion between two given states.
    • “Exact Isospectral Pairs of PT-Symmetric Hamiltonians” by Carl M. Bender, Daniel W. Hook (arXiv:0802.2910 2008.04.27; J.Phys.A41:244005,2008) – http://arxiv.org/abs/0802.2910
      • Abstract: A technique for constructing an infinite tower of pairs of PT-symmetric Hamiltonians, \hat{H}_n and \hat{K}_n (n=2,3,4,…), that have exactly the same eigenvalues is described. The eigenvalue problem for the first Hamiltonian \hat{H}_n of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes’ wedges in the complex plane. The eigenfunctions of the second Hamiltonian \hat{K}_n of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equivalence constitutes a proof that the eigenvalues of both Hamiltonians are real. Although the eigenvalue differential equation associated with \hat{K}_n is real, the Hamiltonian \hat{K}_n exhibits quantum anomalies (terms proportional to powers of \hbar). These anomalies are remnants of the complex nature of the equivalent Hamiltonian \hat{H}_n. In the classical limit in which the anomaly terms in \hat{K}_n are discarded, the pair of Hamiltonians H_{n,classical} and K_{n,classical} have closed classical orbits whose periods are identical.
      • Extracts:
        • Paper considers the first three members of the family of PT-symmetric Hamiltonians \hat{H}_n = \eta \hat{p}^n - \gamma (i\hat{x})^{n^2} for n=2,3,4 and \eta, \gamma \in {\Bbb R}^{+}. Each of these Hamiltonians has the same discrete real spectrum as a corresponding member of another family of Hamiltonians \hat{K}_n which are Hermitian.
        • Example n=2: For \hat{H}_2 = \frac1{2m} \hat{p}^2 - \gamma \hat{x}^{4} one gets \hat{K}_2 = \frac1{2m} \hat{x}^2 + 4 \gamma \hat{p}^{4} + \hbar \sqrt{\frac{2\gamma}{m}} \hat{p}. Notice the quantum-anomaly term (proportional to \hbar). The corresponding classical Hamiltonians H_{2,cl} = \frac1{2m} p^2 - \gamma x^{4} and K_{2,cl} = \frac1{2m} x^2 + 4 \gamma p^{4} also have a correspondence between them: namely, each closed orbit of first one has a corresponding closed orbit of the second one, both orbits with the exactly same periods.
        • Example n=3:
          • \hat{H}_3 = \eta \hat{p}^3 - i \gamma \hat{x}^{9}
          • \hat{K}_3 = i \eta \hat{x}^3 + i (- 27\eta^{\frac23}\gamma^{\frac13}\hbar\hat{p}^2 + 243 \eta^{\frac13}\gamma^{\frac23} \hat{p}^6) \hat{x} + (972 \eta^{\frac13}\gamma^{\frac23}\hbar\hat{p}^5 - 6\eta^{\frac23}\gamma^{\frac13} \hbar^2\hat{p} + 1458 \gamma \hat{p}^9).
          • H_{3,cl} = \eta p^3 - i \gamma x^{9}
          • K_{3,cl} = i \eta x^3 + i 243 \eta^{\frac13}\gamma^{\frac23} p^6 x + 1458 \gamma p^9.
        • Example n=4:
          • \hat{H}_4 = \eta \hat{p}^4 - \gamma \hat{x}^{16}
          • \hat{K}_4 = - \eta \hat{x}^4 + 3 \cdot 2^{24} \gamma \hat{p}^{16} + 2^{18} \gamma^{\frac34}\eta^{\frac14}(8 i \hat{p}^{12} \hat{x} + 54 \hbar \hat{p}^{11}) + 2^{10} \sqrt{\gamma\eta} (-24 \hat{p}^8\hat{x}^2 + i 240 \hbar \hat{p}^7\hat{x} + 483 \hbar^2 \hat{p}^6) - 8 \gamma^{\frac14}\eta^{\frac34} (48 \hbar \hat{p}^3 \hat{x}^2 -i 6 \hbar^2 \hat{p}^2\hat{x} + 87 \hbar^3 \hat{p}).
          • H_{4,cl} = \eta p^4 - \gamma x^{16}
          • K_{4,cl} = - \eta x^4 + 3 \cdot 2^{24} \gamma p^{16} + i 2^{21} \gamma^{\frac34}\eta^{\frac14} p^{12} x - 3 \cdot 2^{13} \sqrt{\gamma\eta} p^8 x^2.
    • “Does the complex deformation of the Riemann equation exhibit shocks?” by Carl M. Bender, Joshua Feinberg (arXiv:0709.2727 2007.09.17; J.Phys.A41:244004,2008) – http://arxiv.org/abs/0709.2727
        Abstract: The Riemann equation u_t+uu_x=0, which describes a one-dimensional accelerationless perfect fluid, possesses solutions that typically develop shocks in a finite time. This equation is PT symmetric. A one-parameter PT-invariant complex deformation of this equation, u_t-iu(iu_x)^\epsilon= 0 (\epsilon real), is solved exactly using the method of characteristic strips, and it is shown that for real initial conditions, shocks cannot develop unless \epsilon is an odd integer.
    • “No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model” by Carl M. Bender, Philip D. Mannheim (arXiv:0706.0207 2007.06.01; Phys.Rev.Lett.100:110402,2008) – http://arxiv.org/abs/0706.0207
        Abstract: Contrary to common belief, it is shown that theories whose field equations are higher than second order in derivatives need not be stricken with ghosts. In particular, the prototypical fourth-order derivative Pais-Uhlenbeck oscillator model is shown to be free of states of negative energy or negative norm. When correctly formulated (as a PT symmetric theory), the theory determines its own Hilbert space and associated positive-definite inner product. In this Hilbert space the model is found to be a fully acceptable quantum-mechanical theory that exhibits unitary time evolution.
    • “Faster than Hermitian Quantum Mechanics” by Carl M. Bender, Dorje C. Brody, Hugh F. Jones, Bernhard K. Meister (arXiv:quant-ph/0609032 2006.09.05; Phys.Rev.Lett.98:040403,2007) – http://arxiv.org/abs/quant-ph/0609032
        Abstract: Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a Hilbert space, there exist Hamiltonians H under which |psi_I> evolves into |psi_F>. Consider the following quantum brachistochrone problem: Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time tau? For Hermitian Hamiltonians tau has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, tau can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from |psi_I> to |psi_F> can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.
    • “Equivalence of a Complex PT-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly” by Carl M. Bender, Dorje C. Brody, Jun-Hua Chen, Hugh F. Jones, Kimball A. Milton, Michael C. Ogilvie (arXiv:hep-th/0605066 2006.05.08; Phys.Rev.D74:025016,2006) – http://arxiv.org/abs/hep-th/0605066
        Abstract: In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian PT-symmetric wrong-sign quartic Hamiltonian H=\frac12 p^2-gx^4 has the same spectrum as the conventional Hermitian Hamiltonian \tilde H=\frac12 p^2+4g x^4-\sqrt{2g} x. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian PT-symmetric Hamiltonian. This anomaly in the Hermitian form of a PT-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into H. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to -\phi^4 quantum field theory in higher-dimensional space-time are discussed.
    • “Calculation of the Hidden Symmetry Operator for a PT-Symmetric Square Well” by Carl M. Bender, Barnabas Tan (arXiv:quant-ph/0601123 2006.01.18; J.Phys.A39:1945-1953,2006) – http://arxiv.org/abs/quant-ph/0601123
        Abstract: It has been shown that a Hamiltonian with an unbroken PT symmetry also possesses a hidden symmetry that is represented by the linear operator C. This symmetry operator C guarantees that the Hamiltonian acts on a Hilbert space with an inner product that is both positive definite and conserved in time, thereby ensuring that the Hamiltonian can be used to define a unitary theory of quantum mechanics. In this paper it is shown how to construct the operator C for the PT-symmetric square well using perturbative techniques.
    • “PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics” by Carl M. Bender, Jun-Hua Chen, Kimball A. Milton (arXiv:hep-th/0511229 2005.11.23; J.Phys.A39:1657-1668,2006) – http://arxiv.org/abs/hep-th/0511229
        Abstract: A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian ix^3 quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green’s function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT-symmetric Hamiltonian. The $\mathcal{O}(g^4)$ contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.
    • “Semiclassical analysis of a complex quartic Hamiltonian” by Carl M. Bender, Dorje C. Brody, Hugh F. Jones (arXiv:quant-ph/0509034 2009.09.05; Phys.Rev.D73:025002,2006) – http://arxiv.org/abs/quant-ph/0509034
        Abstract: It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\frac12 p^2+\frac12\mu^2x^2-\lambda x^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\frac12 p^2+\frac12 \mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter \lambda.
    • “Reflectionless Potentials and PT Symmetry” by Zafar Ahmed, Carl M. Bender, M. V. Berry (arXiv:quant-ph/0508117 2005.08.16; J.Phys.A38:L627-L630,2005) – http://arxiv.org/abs/quant-ph/0508117
        Abstract: Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken PT symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the PT-symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The PT-symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.
    • “Dual PT-Symmetric Quantum Field Theories” by Carl M. Bender, H. F. Jones, R. J. Rivers (arXiv:hep-th/0508105 2005.08.15; Phys.Lett. B625 (2005) 333-340) – http://arxiv.org/abs/hep-th/0508105
        Abstract: Some quantum field theories described by non-Hermitian Hamiltonians are investigated. It is shown that for the case of a free fermion field theory with a \gamma_5 mass term the Hamiltonian is PT-symmetric. Depending on the mass parameter this symmetry may be either broken or unbroken. When the PT symmetry is unbroken, the spectrum of the quantum field theory is real. For the PT-symmetric version of the massive Thirring model in two-dimensional space-time, which is dual to the PT-symmetric scalar Sine-Gordon model, an exact construction of the C operator is given. It is shown that the PT-symmetric massive Thirring and Sine-Gordon models are equivalent to the conventional Hermitian massive Thirring and Sine-Gordon models with appropriately shifted masses.
    • “New Quasi-Exactly Solvable Sextic Polynomial Potentials” by Carl M. Bender, Maria Monou (arXiv:quant-ph/0501053 2005.01.11) – http://arxiv.org/abs/quant-ph/0501053
        Abstract: A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in closed form. An entirely new class of QES Hamiltonians having sextic polynomial potentials is constructed. These new Hamiltonians are different from the sextic QES Hamiltonians in the literature because their eigenfunctions obey PT-symmetric rather than Hermitian boundary conditions. These new Hamiltonians present a novel problem that is not encountered when the Hamiltonian is Hermitian: It is necessary to distinguish between the parametric region of unbroken PT symmetry, in which all of the eigenvalues are real, and the region of broken PT symmetry, in which some of the eigenvalues are complex. The precise location of the boundary between these two regions is determined numerically using extrapolation techniques and analytically using WKB analysis.
    • “Introduction to PT-Symmetric Quantum Theory” by Carl M. Bender (arXiv:quant-ph/0501052 2005.01.11; Contemp.Phys.46:277-292,2005) – http://arxiv.org/abs/quant-ph/0501052
        Abstract: In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving). To express the Hermiticity of a Hamiltonian, one writes H=H^\dagger, where the symbol \dagger denotes the usual Dirac Hermitian conjugation; that is, transpose and complex conjugate. In the past few years it has been recognized that the requirement of Hermiticity, which is often stated as an axiom of quantum mechanics, may be replaced by the less mathematical and more physical requirement of space-time reflection symmetry (PT symmetry) without losing any of the essential physical features of quantum mechanics. Theories defined by non-Hermitian PT-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level. This paper explains how the requirement of Hermiticity can be evaded and discusses the properties of some non-Hermitian PT-symmetric quantum theories.
    • “The C Operator in PT-Symmetric Quantum Theories” by Carl M. Bender, Joachim Brod, Andre Refig, Moritz Reuter (arXiv:quant-ph/0402026 2004.02.03) – http://arxiv.org/abs/quant-ph/0402026
        Abstract: The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry represented by a linear operator called C. Using C it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are positive, and the dynamics is governed by unitary time evolution. After a review of PT-symmetric quantum mechanics, new results are presented here in which the C operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.
    • “Finite-Dimensional PT-Symmetric Hamiltonians” by Carl M. Bender, Peter N. Meisinger, Qinghai Wang (arXiv:quant-ph/0303174; 2003.03.29) – http://arxiv.org/abs/quant-ph/0303174
        Abstract: This paper investigates finite-dimensional representations of PT-symmetric Hamiltonians. In doing so, it clarifies some of the claims made in earlier papers on PT-symmetric quantum mechanics. In particular, it is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the PT symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D>2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional PT-symmetric matrix Hamiltonian.
    • “Quantised Three-Pillar Problem” by Carl M. Bender, Dorje C. Brody, Bernhard K. Meister (arXiv:quant-ph/0302097; 2003.02.12) – http://arxiv.org/abs/quant-ph/0302097
        Abstract: This paper examines the quantum mechanical system that arises when one quantises a classical mechanical configuration described by an underdetermined system of equations. Specifically, we consider the well-known problem in classical mechanics in which a beam is supported by three identical rigid pillars. For this problem it is not possible to calculate uniquely the forces supplied by each pillar. However, if the pillars are replaced by springs, then the forces are uniquely determined. The three-pillar problem and its associated indeterminacy is recovered in the limit as the spring constant tends to infinity. In this paper the spring version of the problem is quantised as a constrained dynamical system. It is then shown that as the spring constant becomes large, the quantum analog of the ambiguity reemerges as a kind of quantum anomaly.
    • “Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics” by Carl M. Bender, Peter N. Meisinger, Qinghai Wang (arXiv:quant-ph/0211166; 2002.11.26) – http://arxiv.org/abs/quant-ph/0211166
        Abstract: In a recent paper it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT-symmetric Hamiltonian H={1\over2}p^2+{1\over2}x^2 +i\epsilon x^3 using perturbative techniques. It is also shown how to construct the operator C for H={1\over2}p^2+{1\over2}x^2-\epsilon x^4 using nonperturbative methods.
    • “All Hermitian Hamiltonians Have Parity” by Carl M. Bender, Peter N. Meisinger, Qinghai Wang (arXiv:quant-ph/0211123; 2002.11.26) – http://arxiv.org/abs/quant-ph/0211123
        Abstract: It is shown that if a Hamiltonian $H$ is Hermitian, then there always exists an operator P having the following properties: (i) P is linear and Hermitian; (ii) P commutes with H; (iii) P^2=1; (iv) the nth eigenstate of H is also an eigenstate of P with eigenvalue (-1)^n. Given these properties, it is appropriate to refer to P as the parity operator and to say that H has parity symmetry, even though P may not refer to spatial reflection. Thus, if the Hamiltonian has the form H=p^2+V(x), where V(x) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has PT symmetry. This shows that PT symmetry is a generalization of Hermiticity: All Hermitian Hamiltonians of the form H=p^2+V(x) have PT symmetry, but not all PT-symmetric Hamiltonians of this form are Hermitian.
    • “Complex Extension of Quantum Mechanics” by Carl M. Bender, Dorje C. Brody, Hugh F. Jones (arXiv:quant-ph/0208076 2002.10.30; EconfC0306234:617-628,2003; Phys.Rev.Lett.89:270401,2002;) – http://arxiv.org/abs/quant-ph/0208076
        Abstract: It is shown that the standard formulation of quantum mechanics in terms of Hermitian Hamiltonians is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct to explain experimental data. One might expect that a quantum theory based on a non-Hermitian Hamiltonian would violate unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed physical symmetry C of the Hamiltonian. Using C, an inner product is constructed whose associated norm is positive definite. This construction is completely general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalisation of it.
    • “Quantum Complex Henon-Heiles Potentials” by Carl M. Bender, Gerald V. Dunne, Peter N. Meisinger, Mehmet Simsek (arXiv:quant-ph/0101095 2001.01.18; Phys.Lett. A281 (2001) 311-316) – http://arxiv.org/abs/quant-ph/0101095
        Abstract: Quantum-mechanical PT-symmetric theories associated with complex cubic potentials such as V=x^2+y^2+igxy^2 and V=x^2+y^2+z^2+igxyz, where g is a real parameter, are investigated. These theories appear to possess real, positive spectra. Low-lying energy levels are calculated to very high order in perturbation theory. The large-order behavior of the perturbation coefficients is determined using multidimensional WKB tunneling techniques. This approach is also applied to the complex Henon-Heiles potential V=x^2+y^2+ig(xy^2-x^3/3).
    • “Variational Ansatz for PT-Symmetric Quantum Mechanics” by Carl Bender, Fred Cooper, Peter Meisinger, Van M. Savage (arXiv:quant-ph/9907008 ; Phys.Lett. A259 (1999) 224-231) – http://arxiv.org/abs/quant-ph/9907008
        Abstract: A variational calculation of the energy levels of a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian H= p^2 - (ix)^N with N positive and x complex is presented. Excellent agreement is obtained for the ground state and low lying excited state energy levels and wave functions. We use an energy functional with a three parameter class of PT-symmetric trial wave functions in obtaining our results.
    • “Complex Square Well — A New Exactly Solvable Quantum Mechanical Model” by Carl M. Bender (Washington U.), Stefan Boettcher (Emory U.), H. F. Jones (Imperial C.), Van M. Savage (Washington U.) (arXiv:quant-ph/9906057; J.Phys.A32:6771-6781,1999) – http://arxiv.org/abs/quant-ph/9906057
        Abstract: Recently, a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian H=p^2+x^2(ix)^\epsilon was studied. It was found that the energy levels for this theory are real for all \epsilon\geq0. Here, the limit as \epsilon\to\infty is examined. It is shown that in this limit, the theory becomes exactly solvable. A generalization of this Hamiltonian, H=p^2+x^{2M}(ix)^\epsilon (M=1,2,3,...) is also studied, and this PT-symmetric Hamiltonian becomes exactly solvable in the large-\epsilon limit as well. In effect, what is obtained in each case is a complex analog of the Hamiltonian for the square well potential. Expansions about the large-\epsilon limit are obtained.
    • “Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian” by Carl M. Bender, Gerald V. Dunne (arXiv:quant-ph/9812039; J.Math.Phys. 40 (1999) 4616-4621) – http://arxiv.org/abs/quant-ph/9812039
        Abstract: A precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian H=p^2+{1/4}x^2+i \lambda x^3, is performed using high-order Rayleigh-Schr\”odinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. The Rayleigh-Schr\”odinger perturbation series is Borel summable, and Pad\’e summation provides excellent agreement with the real energy spectrum. Pad\’e analysis provides strong numerical evidence that the once-subtracted ground-state energy considered as a function of \lambda^2 is a Stieltjes function. The analyticity properties of this Stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable Hamiltonian H=p^2+{1/4}x^2-\epsilon x^3.
    • “PT-Symmetric Quantum Mechanics” by Carl Bender (Washington U.), Stefan Boettcher (Los Alamos and Clark Atlanta U.), Peter Meisinger (Washington U.) (arXiv:quant-ph/9809072; J.Math.Phys. 40 (1999) 2201-2229) – http://arxiv.org/abs/quant-ph/9809072
        Abstract: This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H^\dagger=H on the Hamiltonian, where \dagger represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian $H$ has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H^\ddag=H, where \ddag represents combined parity reflection and time reversal {\cal PT}, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p^2+x^2(ix)^\epsilon of the harmonic oscillator Hamiltonian, where \epsilon is a real parameter. The system exhibits two phases: When \epsilon\geq0, the energy spectrum of H is real and positive as a consequence of {\cal PT} symmetry. However, when -1<\epsilon-N; each of these complex Hamiltonians exhibits a phase transition at \epsilon=0. These {\cal PT}-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
    • “Quasi-exactly solvable quartic potential” by Carl M. Bender, Stefan Boettcher (CNLS, Los Alamos and CTSPS, Clark Atlanta University) (arXiv:physics/9801007; J.Phys. A31 (1998) L273-L277) – http://arxiv.org/abs/physics/9801007
        Abstract: A new two-parameter family of quasi-exactly solvable quartic polynomial potentials V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x is introduced. Until now, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial potential is sextic. This belief is based on the assumption that the Hamiltonian must be Hermitian. However, it has recently been discovered that there are huge classes of non-Hermitian, {\cal PT}-symmetric Hamiltonians whose spectra are real, discrete, and bounded below [physics/9712001]. Replacing Hermiticity by the weaker condition of {\cal PT} symmetry allows for new kinds of quasi-exactly solvable theories. The spectra of this family of quartic potentials discussed here are also real, discrete, and bounded below, and the quasi-exact portion of the spectra consists of the lowest J eigenvalues. These eigenvalues are the roots of a Jth-degree polynomial.
    • “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry” by Carl M. Bender, Stefan Boettcher (CNLS, Los Alamos, and CTSPS, Clark Atlanta University) (arXiv:physics/9712001; Phys.Rev.Lett. 80 (1998) 5243-5246) – http://arxiv.org/abs/physics/9712001
        Abstract: The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of {\cal PT} symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These {\cal PT} symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories.


Fermi paradox

Filed under: astronomy, evolution, life, mind & brain, physics — Tags: , — sandokan65 @ 10:03

Be quiet you stupid young species! (or else be gone)

  • “Is ET avoiding us out of a fear of human galactic conquest?” (io9; 2011.04.10) – http://io9.com/#!5790567/is-et-avoiding-us-out-of-a-fear-of-human-galactic-conquest
  • “Is ET Scared of Human Conquest?” (13point7 blog) – http://thirteenpointseven.wordpress.com/2011/04/11/is-et-scared-of-human-conquest/
  • “Interstellar Predation Could Explain Fermi Paradox” (The Physics arXiv blog; 2011.04.08) – http://www.technologyreview.com/blog/arxiv/26622/
      In a casual chat over lunch back in 1950, the Italian-American physicist Enrico Fermi posed a now famous question. If intelligent life has evolved many times in our galaxy and beyond, why do we see no sign of it?

      Kent puts it like this: “If cosmic habitats are widely enough separated that they are very hard to find, by far the best strategy for a typical species to avoid defeat in such competitions may be to avoid entering them, by being inconspicuous enough that no potential adversary identifies its habitat as valuable.”

      That raises important questions about whether humanity is wise to advertise its existence. Various attempts to send messages to the stars have already been made and many scientists have pointed out that this could be a serious mistake, even a suicidal one.

  • “Too Damned Quiet?” by Adrian Kent (arXiv.org > physics > arXiv:1104.0624; 2011.04.04) – http://arxiv.org/abs/1104.0624
      Abstract: It is often suggested that extraterrestial life sufficiently advanced to be capable of interstellar travel or communication must be rare, since otherwise we would have seen evidence of it by now. This in turn is sometimes taken as indirect evidence for the improbability of life evolving at all in our universe. A couple of other possibilities seem worth considering. One is that life capable of evidencing itself on interstellar scales has evolved in many places but that evolutionary selection, acting on a cosmic scale, tends to extinguish species which conspicuously advertise themselves and their habitats. The other is that — whatever the true situation — intelligent species might reasonably worry about the possible dangers of self-advertisement and hence incline towards discretion. These possibilities are discussed here, and some counter-arguments and complicating factors are also considered.

Inside black holes

Filed under: astronomy, General relativity, physics — Tags: , — sandokan65 @ 09:47
  • “Planets Could Orbit Singularities Inside Black Holes” (The Physics arXiv Blog; 2011.04.11) – http://www.technologyreview.com/blog/arxiv/26626/
      The discovery of stable orbits inside certain kinds of black hole implies that planets and perhaps even life could survive inside these weird objects, says one cosmologist
  • “Is there life inside black holes?” by Vyacheslav I. Dokuchaev (arXiv.org > gr-qc > arXiv:1103.6140; 2011.04.09) – http://arxiv.org/abs/1103.6140
      Abstract: Inside the rotating or charged black holes there are bound periodic planetary orbits, which not coming out nor terminated at the central singularity. The stable periodic orbits inside black holes exist even for photons. We call these bound orbits by the orbits of the third kind, following to Chandrasekhar classification for particle orbits in the black hole gravitational field. It is shown that an existence domain for the third kind orbits is a rather spacious, and so there is a place for life inside the supermassive black holes in the galactic nuclei. The advanced civilizations of the third kind (according to Kardashev classification) may inhabit the interiors of supermassive black holes, being invisible from the outside and basking as in the light of the central singularity and the orbital photons.

Related here: Astronomy & Astrophysics – https://eikonal.wordpress.com/2011/05/09/astronomy-astrophysics/ | Astronomic tables and calculators – https://eikonal.wordpress.com/2010/01/28/astronomic-tables-and-calculators/ | Solar planetary system – https://eikonal.wordpress.com/2011/02/17/solar-planetary-system/ | Physics Sites – https://eikonal.wordpress.com/2010/02/12/physics-sites/


Solar planetary system

Dwarf planets

Definitions (from “Pluto’s Dwarf Planet Family Could Get Bigger” by Ian O’Neill (Discovery News; 2010.04.09) – http://news.discovery.com/space/plutos-dwarf-planet-family-is-about-to-get-bigger.html)

  • The 2006 IAU definition of a dwarf planet states that such a body should be massive enough to achieve hydrostatic equilibrium, it must orbit the sun, but it cannot “clear its own orbit.”
  • The ‘potato-limit’: “the rule that they must be of certain brightness, dwarf planets have a minimum radius of 420 km (260 miles).”

All except Ceres are trans-Neptunians residing in the Kuipert belt.

Name Location Discovery details Orbit radius Orbital period Size (radius) Mass Length of the day Misc
Eris (= 2003 UB_313; initially called “Xena”) Kuipert belt
Pluto Kuipert belt 1930.02.18 by Clyde Tombaugh it has very excentric orbit, with aphelion = 7.375 x10^9 km = 49.305 AU and perihelion = 4.436 x10^9 km = 29.658 AU 90,613.305 days_{Earth} = 248.09 years_{Earth} = 14,164.4 days_{Pluto} 1,153 \pm 10 km = 0.18 R_{Earth} (1.305 \pm 0.007)\times 1022 kg = 0.002  M_{Earth} = 0.178 M_{Moon} 6.387 230 day_{Earth} = 6 \ d \ 9 \ h \ 17 \ m \ 36 \ s Has four satellites:

  • Hydra: discovered by Hubble telescope in 2005. 30 to 115km diameter.
  • Charon: discovered in 1978 by the US Naval Observatory. 1,200km diameter.
  • Nix: discovered by Hubble telescope in 2005. 30 to 115km diameter.
  • P4 (yet unnamed): diameter 13 to 34 km [of 8 to 21 miles]
Makemake Kuipert belt
Haumea Kuipert belt
Quaoar (= 2002 LM60; = 50000 Quaoar) Kuipert belt announced 2002.02.07 by Michael Brown and Chadwick Trujillo of Caltech ~ 6.5 x10^9 km = 4 x10^9 miles ~ 288 years_{Earth} ~ 650 km = 400 miles likely composition: made mostly of low-density ices mixed with rock
Sedna (= 90377 Sedna) Kuipert belt discovered in 2003
Ceres asteroid belt


Oort Cloud

a sphere one light year in radius stretching a quarter of the distance to Alpha Centauri

Potential planet: Tyche

Related here: Astronomy & Astrophysics – https://eikonal.wordpress.com/2011/05/09/astronomy-astrophysics/ | Astronomic tables and calculators – https://eikonal.wordpress.com/2010/01/28/astronomic-tables-and-calculators/ | Inside black holes – https://eikonal.wordpress.com/2011/04/12/inside-black-holes/ | Physics Sites – https://eikonal.wordpress.com/2010/02/12/physics-sites/


Heath kernels

Filed under: eikonal approximation, physics — Tags: , — sandokan65 @ 13:56
  • “Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation” by Matthias Keller, Daniel Lenz (arXiv:1101.2979v1 [math.FA]; 2011.01.15) – http://arxiv.org/abs/1101.2979
      Abstract: We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.
  • “Note on basic features of large time behaviour of heat kernels” by Matthias Keller, Daniel Lenz, Hendrik Vogt, Radosław Wojciechowski (arXiv:1101.0373v1 [math.FA]; 2011.01.11) – http://arxiv.org/abs/1101.0373
      Abstract: Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.

Post-Newtonian gravity

  • Clifford M. Will’s papers on Post-Newtonian approach:
    • 0) “Generation of Post-Newtonian Gravitational Radiation via Direct Integration of the Relaxed Einstein Equations” by Clifford M. Will (arXiv:gr-qc/9910057v1; 1999.10.15) – http://arxiv.org/abs/gr-qc/9910057
        Abstract: The completion of a network of advanced laser-interferometric gravitational-wave observatories around 2001 will make possible the study of the inspiral and coalescence of binary systems of compact objects (neutron stars and black holes), using gravitational radiation. To extract useful information from the waves, such as the masses and spins of the bodies, theoretical general relativistic gravitational waveform templates of extremely high accuracy will be needed for filtering the data, probably as accurate as $O[(v/c)^6]$ beyond the predictions of the quadrupole formula. We summarize a method, called DIRE, for Direct Integration of the Relaxed Einstein Equations, which extends and improves an earlier framework due to Epstein and Wagoner, in which Einstein’s equations are recast as a flat spacetime wave equation with source composed of matter confined to compact regions and gravitational non-linearities extending to infinity. The new method is free of divergences or undefined integrals, correctly predicts all gravitational wave “tail” effects caused by backscatter of the outgoing radiation off the background curved spacetime, and yields radiation that propagates asymptotically along true null cones of the curved spacetime. The method also yields equations of motion through $O[(v/c)^4]$, radiation-reaction terms at $O[(v/c)^5]$ and $O[(v/c)^7]$, and gravitational waveforms and energy flux through $O[(v/c)^4]$, in agreement with other approaches. We report on progress in evaluating the $O[(v/c)^6]$ contributions.
    • 1) “Post-Newtonian Gravitational Radiation and Equations of Motion via Direct Integration of the Relaxed Einstein Equations. I. Foundations” by Michael E. Pati, Clifford M. Will (arXiv:gr-qc/0007087v1; 2000.07.31) – http://arxiv.org/abs/gr-qc/0007087
        Abstract: We present a self-contained framework called Direct Integration of the Relaxed Einstein Equations (DIRE) for calculating equations of motion and gravitational radiation emission for isolated gravitating systems based on the post-Newtonian approximation. We cast the Einstein equations into their “relaxed” form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve the equations formally as a retarded integral over the past null cone of the field point (chosen to be within the near zone when calculating equations of motion, and in the far zone when calculating gravitational radiation). The “inner” part of this integral(within a sphere of radius {\cal R} \sim one gravitational wavelength) is approximated in a slow-motion expansion using standard techniques; the “outer” part, extending over the radiation zone, is evaluated using a null integration variable. We show generally and explicitly that all contributions to the inner integrals that depend on {\cal R} cancel corresponding terms from the outer integrals, and that the outer integrals converge at infinity, subject only to reasonable assumptions about the past behavior of the source. The method cures defects that plagued previous “brute-force” slow-motion approaches to motion and gravitational radiation for isolated systems. We detail the procedure for iterating the solutions in a weak-field, slow-motion approximation, and derive expressions for the near-zone field through 3.5 post-Newtonian order in terms of Poisson-like potentials.
    • 2) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. II. Two-body equations of motion to second post-Newtonian order, and radiation-reaction to 3.5 post-Newtonian order” by Michael E. Pati, Clifford M. Will (arXiv:gr-qc/0201001v1; 2001.12.31) – http://arxiv.org/abs/gr-qc/0201001
        Abstract: We derive the equations of motion for binary systems of compact bodies in the post-Newtonian (PN) approximation to general relativity. Results are given through 2PN order (order (v/c)^4 beyond Newtonian theory), and for gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method is based on a framework for direct integration of the relaxed Einstein equations (DIRE) developed earlier, in which the equations of motion through 3.5PN order can be expressed in terms of Poisson-like potentials that are generalizations of the instantaneous Newtonian gravitational potential, and in terms of multipole moments of the system and their time derivatives. All potentials are well defined and free of divergences associated with integrating quantities over all space. Using a model of the bodies as spherical, non-rotating fluid balls whose characteristic size s is small compared to the bodies’ separation r, we develop a method for carefully extracting only terms that are independent of the parameter s, thereby ignoring tidal interactions, spin effects, and internal self-gravity effects. Through 2.5PN order, the resulting equations agree completely with those obtained by other methods; the new 3.5PN back-reaction results are shown to be consistent with the loss of energy and angular momentum via radiation to infinity.
    • 3) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. III. Radiation reaction for binary systems with spinning bodies” by Clifford M. Will (arXiv:gr-qc/0502039v2; 2005.04.29) – http://arxiv.org/abs/gr-qc/0502039
        Abstract: Using post-Newtonian equations of motion for fluid bodies that include radiation-reaction terms at 2.5 and 3.5 post-Newtonian (PN) order (O[(v/c)^5] and O[(v/c)^7] beyond Newtonian order), we derive the equations of motion for binary systems with spinning bodies. In particular we determine the effects of radiation-reaction coupled to spin-orbit effects on the two-body equations of motion, and on the evolution of the spins. For a suitable definition of spin, we reproduce the standard equations of motion and spin-precession at the first post-Newtonian order. At 3.5PN order, we determine the spin-orbit induced reaction effects on the orbital motion, but we find that radiation damping has no effect on either the magnitude or the direction of the spins. Using the equations of motion, we find that the loss of total energy and total angular momentum induced by spin-orbit effects precisely balances the radiative flux of those quantities calculated by Kidder et al. The equations of motion may be useful for evolving inspiraling orbits of compact spinning binaries.
    • 4) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. IV. Radiation reaction for binary systems with spin-spin coupling” by Han Wang, Clifford M. Will (arXiv:gr-qc/0701047v2; 2007.03.16) – http://arxiv.org/abs/gr-qc/0701047
        Abstract: Using post-Newtonian equations of motion for fluid bodies that include radiation-reaction terms at 2.5 and 3.5 post-Newtonian (PN) order O[(v/c)^5] and O[(v/c)^7] beyond Newtonian order), we derive the equations of motion for binary systems with spinning bodies, including spin-spin effects. In particular we determine the effects of radiation-reaction coupled to spin-spin effects on the two-body equations of motion, and on the evolution of the spins. We find that radiation damping causes a 3.5PN order, spin-spin induced precession of the individual spins. This contrasts with the case of spin-orbit coupling, where there is no effect on the spins at 3.5PN order. Employing the equations of motion and of spin precession, we verify that the loss of total energy and total angular momentum induced by spin-spin effects precisely balances the radiative flux of those quantities calculated by Kidder et al.
    • 5) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. V. Evidence for the strong equivalence principle to second post-Newtonian order” by Thomas Mitchell, Clifford M. Will (arXiv:0704.2243v2 [gr-qc]; 2007.07.17)- http://arxiv.org/abs/0704.2243
        Abstract: Using post-Newtonian equations of motion for fluid bodies valid to the second post-Newtonian order, we derive the equations of motion for binary systems with finite-sized, non-spinning but arbitrarily shaped bodies. In particular we study the contributions of the internal structure of the bodies (such as self-gravity) that would diverge if the size of the bodies were to shrink to zero. Using a set of virial relations accurate to the first post-Newtonian order that reflect the stationarity of each body, and redefining the masses to include 1PN and 2PN self-gravity terms, we demonstrate the complete cancellation of a class of potentially divergent, structure-dependent terms that scale as s^{-1} and s^{-5/2}, where s is the characteristic size of the bodies. This is further evidence of the Strong Equivalence Principle, and supports the use of post-Newtonian approximations to derive equations of motion for strong-field bodies such as neutron stars and black holes. This extends earlier work done by Kopeikin.


Vacuum energy

Filed under: physics, qft — Tags: , , — sandokan65 @ 22:57


Anthropic principle

  • “Evidence Against Fine Tuning for Life” by Don N. Page (arXiv:1101.2444; 2011.01.12) – http://arxiv.org/abs/1101.2444
      Abstract: The effective coupling `constants’ of physics, especially the cosmological constant, are observed to have highly biophilic values. If this is not a hugely improbable accident, or a consequence of some mysterious logical necessity or of some simple principle of physics, it might be explained as a consequence either of an observership selection principle within a multiverse of many sets of effective coupling constants, or else of some biophilic principle that fine tunes the constants of physics to optimize life. Here evidence is presented against the hypothesis of fine tuning by a biophilic principle that maximizes the fraction of baryons that form living beings.
  • Anthropic principle (WikiPedia) – http://en.wikipedia.org/wiki/Anthropic_principle
  • Fine-tuned Universe (WikiPedia) – http://en.wikipedia.org/wiki/Fine-tuned_Universe
  • Cosmological constant (WikiPedia) – http://en.wikipedia.org/wiki/Cosmological_constant



Filed under: mathematics, physics — Tags: — sandokan65 @ 14:33



Exact solutions

Filed under: physics, qft — Tags: — sandokan65 @ 02:55
  • “Mass generation and supersymmetry” by Marco Frasca (arXiv:1007.5275; 2010.12.26) – http://arxiv.org/abs/1007.5275
      Abstract: Using a recent understanding of mass generation for Yang-Mills theory and a quartic massless scalar field theory mapping each other, we show that when such a scalar field theory is coupled to a gauge field and Dirac spinors, all fields become massive at a classical level with all the properties of supersymmetry fulfilled, when the self-interaction of the scalar field is taken infinitely large. Assuming that the mechanism for mass generation must be the same in QCD as in the Standard Model, this implies that Higgs particle must be supersymmetric.
  • “Exact solutions of classical scalar field equations” by Marco Frasca (arXiv:0907.4053; 2009.07.23) – http://arxiv.org/abs/0907.4053
      Abstract: We give a class of exact solutions of quartic scalar field theories. These solutions prove to be interesting as are characterized by the production of mass contributions arising from the nonlinear terms while maintaining a wave-like behavior. So, a quartic massless equation has a nonlinear wave solution with a dispersion relation of a massive wave and a quartic scalar theory gets its mass term renormalized in the dispersion relation through a term depending on the coupling and an integration constant. When spontaneous breaking of symmetry is considered, such wave-like solutions show how a mass term with the wrong sign and the nonlinearity give rise to a proper dispersion relation. These latter solutions do not change the sign maintaining the property of the selected value of the equilibrium state. Then, we use these solutions to obtain a quantum field theory for the case of a quartic massless field. We get the propagator from a first order correction showing that is consistent in the limit of a very large coupling. The spectrum of a massless quartic scalar field theory is then provided. From this we can conclude that, for an infinite countable number of exact classical solutions, there exist an infinite number of equivalent quantum field theories that are trivial in the limit of the coupling going to infinity.


List of historic physics papers available on the web



More papers at http://dieumsnh.qfb.umich.mx/archivoshistoricosMQ/:


Fluid dynamics

Filed under: complexity, fluid dynamics, physics — Tags: — sandokan65 @ 11:49


“Quantum” calculus

Currently all this material is retyped from the reference [1].


  • The q-analogue of n: [n]:\equiv \frac{q^n-1}{q-1};
  • The q-analogue of factorial n!: [n]! :\equiv \prod_{k=1}^{n}[k] for k\in{\Bbb N} (and [0]!=1);
  • (x-a)^{n}_{q} :\equiv \prod_{k=0}^{n-1} (x-q^k a) for n\in{\Bbb N}.
  • (x-a)^{-n}_{q} :\equiv  \frac1{(x- q^{-n}a)^{n}_{q}}.


  • [-n] = - q^{-n} [n].
  • (x-a)^{m+n}_{q} =  (x-a)^{m}_{q} (x- q^m a)^{n}_{q},
  • (a-x)^{n}_{q} = (-)^n q^{n(n-1)/2} (x - q^{-n+1}a)^{n}_{q},

the “quantum” differentials

Definitions: For an arbitrary function f:{\Bbb R}\rightarrow {\Bbb R} define:

  • its q-differential: d_q f(x) :\equiv f(q x) - f(x);
  • its h-differential: d_h f(x) :\equiv f(x+h) - f(x);
  • its q-derivative: D_q f(x) :\equiv \frac{d_q f(x)}{d_q x} = \frac{f(q x) - f(x)}{(q-1)x};
  • its h-derivative: D_h f(x) :\equiv \frac{d_h f(x)}{d_h x} = \frac{f(x+h) - f(x)}{h};

Note that d_q x = (q-1) x and d_h x = h.

Basic properties:

  • all four operators (d_q, d_h, D_q and D_h) are linear, e.g. d_q (\alpha f(x) + \beta g(x)) = \alpha d_q f(x) + \beta d_q g(x).
  • d_q (f(x)g(x)) = (d_q f(x)) g(x) + f(qx) (d_q g(x));
  • d_h (f(x)g(x)) = (d_h f(x)) g(x) + f(x+h) (d_h g(x));
  • D_q (f(x)g(x)) = (D_q f(x)) g(x) + f(qx) (D_q g(x));
  • D_q \left(\frac{f(x)}{g(x)}\right) = \frac{D_q f(x) \ g(x) - f(x)  \ D_q g(x)}{g(x)g(qx)} = \frac{D_q f(x) \ g(q x) - f(q x) \ D_q g(x)}{g(x)g(qx)} ;
  • there does not exist a general chain rule for q-derivatives
  • such rule exists for monomial changes of variables x\rightarrow x' = \alpha x^{\beta}, where D_q f(x'(x)) = (D_{q^\beta}f)(x') \cdot D_q x'(x).

Examples and properties:

  • D_q x^n = [n] x^{n-1}.
  • (D^n_q f)(0) = \frac{f^{(n)}(0)}{n!} [n]!.
  • D_q f(x) = \sum_{n=0}^{\infty}\frac{(q-1)^n}{(n+1)!} x^n f^{(n+1)}(x).
  • P_n(x) :\equiv \frac{x^n}{[n]!} satisfies D_q P_n(x) = P_{n-1}(x).
  • D_q (x-a)^{n}_{q} = [n] (x-a)^{n-1}_{q},
  • D_q (a-x)^{n}_{q} = - [n] (a- q x)^{n-1}_{q},
  • D_q \frac1{(x-a)^{n}_{q}} = [-n] (x-q^n a)^{-n-1}_{q},
  • D_q \frac1{(a-x)^{n}_{q}} = \frac{[n]}{(a-x)^{n+1}_{q}},

q-binomial calculus

Definition: The q-binomial coefficient is defined by \left[{n \atop j}\right] :\equiv \frac{[n]!}{[j]![n-j]!}.

Properties of q-binomial coefficients:

  • \left[{n \atop n-j}\right] = \left[{n \atop j}\right].
  • there exist two q-Pascal rules: \left[{n \atop j}\right] = \left[{n-1 \atop j-1}\right] + q^j \left[{n-1 \atop j}\right] and \left[{n \atop j}\right] = q ^{n-j}\left[{n-1 \atop j-1}\right] + \left[{n-1 \atop j}\right].
  • \left[{n \atop 0}\right] = \left[{n \atop n}\right] = 1.
  • \left[{n \atop j}\right] is a polynomial in q of degree j(n-j) with the leading coefficient equal to 1.
  • \left[{\alpha \atop j}\right] = \frac{[\alpha] [\alpha-1] \cdots [\alpha -j +1]}{[j]!} for any number \alpha.
  • \left[{m+n \atop k}\right] = \sum_{j=0}^{k} q^{(k-j)(m-j)} \left[{m \atop j}\right] \left[{n \atop k-j}\right].
  • x^n = \sum_{j=0}^{n} \left[{n \atop j}\right] (x-1)^j_q.
  • \sum_{j=0}^{2m} (-)^j \left[{2m \atop j}\right] = (1-q^{2m-1})(1-q^{2m-3})\cdots(1-q).
  • \sum_{j=0}^{2m+1} (-)^j \left[{2m+1 \atop j}\right] = 0.
  • The Gauss’s binomial formula: (x+a)^n_q = \sum_{j=0}^n \left[{n \atop j}\right] q^{j(j-1)/2} a^{j} x^{n-j}.
  • For two non-commutative operators \hat{A} and \hat{B} s/t \hat{B}\hat{A} = q\hat{A}\hat{B} (with q and ordinary number), the non-commutative Gauss’s binomial formula is: (\hat{A}+\hat{B})^n = \sum_{j=0}^n \left[{n \atop j}\right] \hat{A}^{j} \hat{B}^{n-j}. Such two operators are \hat{x} and \hat{M_q} defined as \hat{x} f(x) :\equiv x f(x) and \hat{M_q} f(x) :\equiv f(qx).
  • The Heine’s binomial formula: \frac1{(1-x)^n_q} = 1+ \sum_{j=1}^{\infty} \frac{[n][n+1]\cdots[n+j-1]}{[j]!} x^j.
  • \frac1{(1-x)^{\infty}_q} = \sum_{j=0}^{\infty}  \frac{x^j}{(1-q)(1-q^2)\cdots(1-q^j)}.
  • (1+x)^{\infty}_q = \sum_{j=0}^{\infty} q^{j(j-1)/2} \frac{x^j}{(1-q)(1-q^2)\cdots(1-q^j)}.

Generalized Taylor’s formula for polynomials

For given number a and linear operator D on space of polynomials, there exist a unique sequence of polynomials \{P_0(x), P_1(x), \cdots\} such that

  • P_0(a)=1 and P_{n>0}(a)=0;
  • \hbox{deg}P_n = n;
  • D P_n(x) = P_{n-1}(x) (\forall n\ge 1) and D(1)=0.

Then any polynomial f(x) of degree n has the unique expansion via following generalized Taylor expression: f(x) = \sum_{j=0}^n (D^j f)(a) P_j(x).


  • If D is D_q we have: f(x) = \sum_{j=0}^n (D_q^j f)(a) \frac{(x-a)^j_q}{[j]!}.
  • for f(x)=x^n and a=1 one gets: x^n = \sum_{j=0}^n \left[{n \atop j}\right] (x-1)^j_q.

Exponentials and trigonometric functions

Definitions: (q-exponentials):

  • e_q^x :\equiv \sum_{k=0}^\infty \frac{x^k}{[k]!} = \frac1{(1-(1-q)x)_q^\infty};
  • E_q^x :\equiv \sum_{k=0}^\infty q^{k(k-1)/2} \frac{x^k}{[k]!} = (1+(1-q)x)_q^\infty.


  • e_q^0 = 1, E_q^0 =1.
  • \frac1{(1-x)_0^\infty} = e_q^{x/(1-q)}.
  • D_q e_q^x = e_q^x, D_q E_q^x = E_q^{qx}.
  • D_q \frac1{(1-(1-q)x)_q^n} = \frac{(1-q)[n]}{(1-(1-q)x)_q^{n+1}}, D_q (1+(1-q)x)_q^n = (1-q) [n] (1+q(1-q)x)^{n-1}_q.
  • e_q^{x}e_q^{y} \ne e_q^{x+y}; but e_q^{x}e_q^{y} = e_q^{x+y} iff yx=qxy.
  • E_q^{-x} = \frac1{e_q^{x}}, i.e. E_q^{x} = \frac1{e_q^{-x}}.
  • e_{\frac1{q}}^{x} = E_q^x.
  • direct consequence of the previous two lines: e_q^{-x}e_{\frac1{q}}^{x} = 1.

Definitions: (q-trigonometric functions):

  • sin_q(x) :\equiv \frac{e_q^{ix}-e_q^{-ix}}{2i},
  • Sin_q(x) :\equiv \frac{E_q^{ix}-E_q^{-ix}}{2i},
  • cos_q(x) :\equiv \frac{e_q^{ix}+e_q^{-ix}}{2i},
  • Cos_q(x) :\equiv \frac{E_q^{ix}+E_q^{-ix}}{2i}.


  • cos_q(x) Cos_q(x) + sin_q(x) Sin_q(x) = 1.
  • D_q sin_q(x) = cos_q(x),
  • D_q Sin_q(x) = Cos_q(qx),
  • D_q cos_q(x) = - sin_q(x),
  • D_q Cos_q(x) = - Sin_q(qx).

Partition functions and product formulas


  • The triangular numbers: \Delta_n :\equiv \frac{n(n+1)}2,
  • the square numbers: \Box_n :\equiv n^2,
  • the pentagonal numbers: \Pi_n :\equiv \frac{n(3n-1)}2,
  • the k-gonal numbers: m^{(k)}_n :\equiv (k-2) \Delta_{n-1} + n =  \frac{n(kn -2n -k +4)}2.

Definition: The classical partition function p(n):{\Bbb Z}\rightarrow{\Bbb N}) is defined as

  • p(n) = the number of ways to partition an positive integer number n into sum of positive integers (modulo reordering of summands);
  • p(n)=0 for n<0;
  • p(0)=1.


  • Examples: p(1)=1, p(2)=2, p(3) =3, p(4) = 5, p(5)=7.
  • Asymptotic behavior: p(n) \sim \frac1{3\sqrt{3}n} e^{\pi \sqrt{\frac{2n}{3}}} \ as \ n\rightarrow \infty.
  • \varphi(q)^{-1} = \sum_{n=0}^\infty p(n) q^n.
  • p(n) = \sum_{n=0}^\infty (-)^{n-1} (p(n-\Pi_n) + p(n-\Pi_{-n})).

Definition: the Euler’s product: \varphi(q) :\equiv  \prod_{n=1}^\infty (1-q^{n}).

Theorem (Jacobi’s triple product identity): For |q|<1 following is true:

    \sum_{n\in{\Bbb Z}} q^{n^2}z^n = \prod_{n=1}^\infty (1-q^{2n})(1+q^{2n-1}z)(1+q^{2n-1}z^{-1}).


  • Euler’s product formula: \sum_{n\in{\Bbb Z}} (-)^n q^{\frac{n(3n-1)}2} =  \prod_{n=1}^\infty (1-q^{n}).
    • This can be rephrased as follows: \varphi(q) = \sum_{n\in{\Bbb Z}} (-)^n q^{\Pi_n} where \Pi_n are the pentagonal numbers defined above.
  • Following Gauss identities are special cases of the Jacobi’s triple product identity:
    • \sum_{n=0}^{\infty} q^{\Delta_n} = \prod_{n=1}^\infty \frac{1-q^{2n}}{1-q^{2n-1}},
    • \sum_{n=0}^{\infty} (-q)^{\Box_n} = \prod_{n=1}^\infty \frac{1-q^{n}}{1+q^{n}}.


  • [1] book: “Quantum Calculus” by Victor Kac and Pokman Cheung (Springer) – ISBN 0-387-9534198; QA303.C537 2001

Other references:


Runge-Lenz vector

For a potential U(\vec{r}) = \frac{k}{r}, the conserved vectors are (slightly changed definitions from reference [1]):

  • the angular momentum verctor: \vec{L} :\equiv \vec{r}\times\vec{p}, and
  • the (dimensionless) eccentricity vector: \vec{e} = \frac{\underline{A}}{mk} :\equiv \frac1{mk}\vec{p}\times\vec{L} - \hat{r} where \hat{r} :\equiv \frac1{r} \vec{r}, and \underline{A} is the LRL vector (the Laplace-Runge-Lenz vector = the Runge–Lenz vector = the Lenz vector).
    • As a consequence, the (dimensionless) vector of binormal is also preserved: \vec{b} :\equiv \vec{l}\times\vec{e} = \frac{L}{mk}\vec{p} +\hat{r}\times\vec{l}. Here \vec{l}:\equiv \frac1{L}\vec{L} =  \vec{e}\times\vec{b}.

  • [1] Wikipedia: Laplace-Runge-Lenz vector – http://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector | Eccentricity vector – http://en.wikipedia.org/wiki/Eccentricity_vector
  • Laplace-Runge-Lenz Vector – http://scienceworld.wolfram.com/physics/Laplace-Runge-LenzVector.html
  • Laplace-Runge-Lenz vector (The Tangent Bundle Physics Wiki) – http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/Laplace-Runge-Lenz_vector
  • P.E.S. Wormer (2003). Properties of the quantum mechanical Runge-Lenz vector. – <a href=http://www.theochem.ru.nl/~pwormer/rungelenz.pdf".http://www.theochem.ru.nl/~pwormer/rungelenz.pdf
  • “Mysteries of the gravitational 2-body problem” by John Baez (2003.05.03) – http://math.ucr.edu/home/baez/gravitational.html
  • “A planar Runge-Lenz vector” by S.G.Kamath (arXiv:hep-th/0112067v1; 2001.12.10; also J.Math.Phys. 43 (2002) 318-324) – http://arxiv.org/abs/hep-th/0112067
      Abstract: Following Dahl’s method an exact Runge-Lenz vector M with two components M and M is obtained as a constant of motion for a two particle-system with charges e and e whose electromagnetic interaction is based on Chern-Simons electrodynamics. The Poisson bracket {M, M} = L but is modified by the appearance of the product e e as central charges.
  • “Duality of force laws and Conformal transformations” by Dawood Kothawala
    (arXiv:1010.2238v3 [physics.class-ph]; 2011.01.07) – http://arxiv.org/abs/1010.2238

      Abstract: As was first noted by Isaac Newton, the two most famous ellipses of classical mechanics, arising out of the force laws F~r and F~1/r^2, can be mapped onto each other by changing the location of center-of-force. What is perhaps less well known is that this mapping can also be achieved by the complex transformation, z -> z^2. We give a simple derivation of this result (and its generalization) by writing the Gaussian curvature in its “covariant” form, and then changing the \emph{metric} by a conformal transformation which “mimics” this mapping of the curves. The final result also yields a relationship between Newton’s constant G, mass M of the central attracting body in Newton’s law, the energy E of the Hooke’s law orbit, and the angular momenta of the two orbits. We also indicate how the conserved Laplace-Runge-Lenz vector for the 1/r^2 force law transforms under this transformation, and compare it with the corresponding quantities for the linear force law. Our main aim is to present this duality in a geometric fashion, by introducing elementary notions from differential geometry.
  • “Laplace-Runge-Lenz symmetry in general rotationally symmetric systems” by Uri Ben-Ya’acov (arXiv:1005.1817; 2010.05.11) – http://arxiv.org/abs/1005.1817
      Abstract: The Laplace-Runge-Lenz symmetry, well known to exist in classical two-body Kepler-Coulomb systems, is also known to be generalizable to all rotationally symmetric systems. It also appears in the computation of the Lorentz boost in relativistic systems. Towards the unification of these properties, the generic (independent of the interaction) properties of the symmetry are verified and extended. The independence of the symmetry on the type of interaction is proven applying only the most minimal properties of the Poisson brackets. Generalized Laplace-Runge-Lenz vectors are definable to be constant (not only piece-wise conserved) for all cases, including open orbits. Also discussed are the transformations generated by the Laplace-Runge-Lenz vectors, the emergence of these vectors in post-Newtonian extensions of general centrally symmetric systems, and the application of these results to relativistic Coulomb systems.
  • “Generalized Laplace-Runge-Lenz vector for the three-dimensional classical motions generated by central forces with a monopole” by T. Yoshida (Il Nuovo Cimento B (1971-1996); Volume 104, Number 4, 375-385, DOI: 10.1007/BF02725670) – http://www.springerlink.com/content/k481322u42ln7244/ [REQUIRES PAYMENT FOR ACCESS]
      Abstract: This paper is concerned with the general method of constructing conserved vectors, which is applicable to noncentral force problems. The point of this method is to express an orthonormal system of vectors in terms of the position, momentum and angular-momentum vectors. Then conserved vectors, such as the Laplace-Runge-Lenz vector, are obtained as a unit vector of the fixed orthonormal system, if the equations of motion are solvable by the quadrature. To facilitate the procedure for obtaining the conserved vectors a unit dyadic is introduced. A generalized Laplace-Runge-Lenz vector is obtained indeed by this procedure for the three-dimensional classical motions generated by central forces with Dirac’s monopole.
  • “Energy spectrum of the two-dimensionalq-hydrogen atom” by Shengli Zhang (International Journal of Theoretical Physics; Volume 34, Number 11, 2217-2221, DOI: 10.1007/BF00673837) – http://www.springerlink.com/content/n251256l88572074/ [REQUIRES PAYMENT FOR ACCESS]
      Abstract: The discrete energy spectrum of theq-analog of the two-dimensional hydrogen atom is derived by deforming the Pauli equation. It contracts to that of the ordinary two-dimensional hydrogen atom in the limitq rarr ± 1. The degeneracy is discussed generally and some properties of theq-energy spectrum are studied both for realq and for complexq of magnitude unity.
  • “The hidden symmetry of the Coulomb problem in relativistic quantum mechanics: From Pauli to Dirac” by Tamari T. Khachidze and Anzor A. Khelashvili (American Journal of Physics — July 2006 — Volume 74, Issue 7, pp. 628-632) – http://ajp.aapt.org/resource/1/ajpias/v74/i7/p628_s1 [REQUIRES PAYMENT FOR ACCESS]
      Abstract: Additional conserved quantities associated with an extra symmetry govern a wide variety of physical systems ranging from planetary motion to atomic spectra. We give a simple derivation of the hidden symmetry operator for the Dirac equation in a Coulomb field and show that this operator may be reduced to the one introduced by Johnson and Lippmann to include the spin degrees of freedom. This operator has been rarely discussed in the literature and has been neglected in recent textbooks on relativistic quantum mechanics and quantum electrodynamics.
  • “Rotating Laplace-Runge-Lenz vector leading to two relativistic Kepler’s equations” by Takeshi Yoshida (Phys. Rev. A 38, 19–25 (1988) = Phys. Rev. A » Volume 38 » Issue 1) – http://pra.aps.org/abstract/PRA/v38/i1/p19_1 [just abstract]
      Abstract: For the nonrelativistic Kepler problem it is well known that there exists a transcendental equation (the so-called Kepler’s equation) which gives a position in the orbit at a given time. The precessing orbit of the relativistic Kepler problem is reduced to the nonprecessing one by using the transformation related to a rotating Laplace-Runge-Lenz vector. This vector is an extension of the familiar conserved vector and is always oriented to the moving perihelion point of the precessing orbit. From the considerations on Kepler’s equation and the rotating Laplace-Runge-Lenz vector, a relativistic Kepler’s equation is newly defined to find a position in the precessing orbit at a given time. This equation has two expressions corresponding to the relativistic coordinate time and proper time. By using the equation, Lambert’s theorem that gives the required time on the orbit is extended to the relativistic problem.
  • “Dynamics of the Laplace-Runge-Lenz vector in the quantum-corrected Newton gravity” by C. Farina, W. J. M. Kort-Kamp, Sebastiao Mauro Filho, Ilya L. Shapiro (arXiv:1101.5611v2 [gr-qc]; 2011.02.02) – http://arxiv.org/abs/1101.5611
      Abstract: Recently it was shown that quantum corrections to the Newton potential can explain the rotation curves in spiral galaxies without introducing the Dark Matter halo. The unique phenomenological parameter $\al\nu$ of the theory grows with the mass of the galaxy. In order to better investigate the mass-dependence of $\al\nu$ one needs to check the upper bound for $\al\nu$ at a smaller scale. Here we perform the corresponding calculation by analyzing the dynamics of the Laplace-Runge-Lenz vector. The resulting limitation on quantum corrections is quite severe, suggesting a strong mass-dependence of $\al\nu$.
  • “Orbit Determination with the two-body Integrals” by Giovanni Federico Gronchi, Linda Dimare, Andrea Milani (arXiv:0911.3555v2 [math-ph]; 2010.03.31) – http://arxiv.org/abs/0911.3555
      Abstract: We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.
  • “Orbit Determination with the two-body Integrals. II” by Giovanni F. Gronchi, Davide Farnocchia, Linda Dimare (arXiv:1101.4569v1 [math-ph]; 2011.01.24) – http://arxiv.org/abs/1101.4569
      Abstract: The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for this problem, that we can solve using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. In this paper we continue the research started in [Gronchi, Dimare, Milani, ‘Orbit determination with the two-body intergrals’, CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. A suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.
  • “The Universal Kepler Problem” by Guowu Meng (arXiv:1011.6609v3 [math-ph]; 2010.12.22) – http://arxiv.org/abs/1011.6609>
      Abstract: For each simple euclidean Jordan algebra, the analogues of hamiltonian, angular momentum and Lenz vector in the Kepler problem are introduced. The analogue of hidden symmetry algebra generated by hamiltonian, angular momentum and Lenz vector is also derived. Finally, for the simplest simple euclidean Jordan algebra, i.e., $\bb R$, we demonstrate how to get generalized Kepler problems by combining with the quantizations of the TKK algebra.
  • pages 90-95 of book “Hamiltonian dynamics” by Gaetano Vilasi – http://tinyurl.com/648t4zf
  • “Determination of the Runge—Lenz Vector” by W. H. Heintz (1974) – http://www.phys.ufl.edu/~maslov/classmech/heinz.pdf [PDF]


Feynman derivation of Maxwell equations

  • “On Feynman’s Approach to the Foundations of Gauge Theory” by M. C. Land, N. M. Shnerb and L. P. Horwitz; 2 Aug 1993. – 36 p. – http://cdsweb.cern.ch/record/568394/:
      Abstract: In 1948, Feynman showed Dyson how the Lorentz force and Maxwell equations could be derived from commutation relations coordinates and velocities. Several authors noted that the derived equations are not Lorentz covariant and so are not the standard Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman’s derivation is of Newtonian form. Tanimura generalized Feynman’s derivation to a Lorentz covariant form, however, this derivation does not lead to the standard Maxwell theory either. Tanimura’s force equation depends on a fifth ({\it scalar}) electromagnetic potential, and the invariant evolution parameter cannot be consistently identified with the proper time of the particle motion. Moreover, the derivation cannot be made reparameterization invariant; the scalar potential causes violations of the mass-shell constraint which this invariance should guarantee. In this paper, we examine Tanimura’s derivation in the framework of the proper time method in relativistic mechanics, and use the technique of Hojman and Shepley to study the unconstrained commutation relations. We show that Tanimura’s result then corresponds to the five-dimensional electromagnetic theory previously derived from a Stueckelberg-type quantum theory in which one gauges the invariant parameter in the proper time method. This theory provides the final step in Feynman’s program of deriving the Maxwell theory from commutation relations; the Maxwell theory emerges as the “correlation limit” of a more general gauge theory, in which it is properly contained.


Uses of Eikonal approximation

Filed under: eikonal approximation, mathematics, physics — Tags: — sandokan65 @ 14:41
  • “Rytov/Eikonal Approximation of Wavepaths” by William S. Harlan (1998.04) – http://billharlan.com/pub/papers/rytov/rytov.html
  • “Structure of entropy solutions to the eikonal equation” by Camillo De Lellis and Felix Otto (Journal: J. Eur. Math. Soc.; Volume: 5; Number: 2; Pages: 107-145; 2003) – http://cvgmt.sns.it/papers/delott02/
      Abstract: In this paper, we establish rectifiability of the jump set of an S1-valued conservation law in two space-dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow-ups.
      The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV-control, which is not available in these variationally motivated problems.
Older Posts »

Create a free website or blog at WordPress.com.