Eikonal Blog


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.


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.


“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:


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.


Harmonic oscillator

Filed under: Quantum mechanics — Tags: , , — sandokan65 @ 20:23

Potential function: U(x)=\frac{m\omega^2 x^2}{2}

Solutions of the Schr\”odinger equation:

  • States: \psi_n(x) = (\frac{m}{\hbar\omega})^{\frac14}  \frac{1}{\sqrt{2^n n! \pi}} H_n(\sqrt{\frac{m\omega}{\hbar}}x) e^{-\frac{m\omega x^2}{\hbar}}
  • Energies: E_n=\hbar\omega (n+\frac12)

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