# Eikonal Blog

## 2011.06.03

### Decoherence

• “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.

## 2011.02.11

### Vacuum energy

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

## 2011.01.02

### 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.

## 2010.07.13

### Time inversion

Filed under: qft — Tags: , , — sandokan65 @ 14:02

Time inversion

• ${\cal T} = {\cal U}{\cal K}$
• $\hat{x}:\equiv (-x^0,\vec{x}) = -\bar{x}$
• ${\cal T}H{\cal T}^{-1} = H$
• ${\cal T}{\cal L}(x){\cal T}^{-1} = {\cal L}(\hat{x})$
• ${\cal T}j_\mu(x){\cal T}^{-1} = j^\mu(\hat{x})$

## Scalar field – real

• ${\cal T}\phi(x){\cal T}^{-1} = \epsilon\phi(\hat{x})$
• ${\cal T}a(\vec{k}){\cal T}^{-1} = \epsilon a(-\vec{k})$
• ${\cal T}a^\dagger(\vec{k}){\cal T}^{-1} = \epsilon a^\dagger(-\vec{k})$
• ${\cal U}a(\vec{k}){\cal U}^{-1} = \epsilon a^\dagger(-\vec{k})$
• ${\cal U}a^\dagger(\vec{k}){\cal U}^{-1} = \epsilon a(-\vec{k})$

where $\epsilon=\pm 1$.

So:
${\cal U} = e^{-\frac{i\pi}2 \int d^3\vec{k} (a^\dagger(\vec{k})a(\vec{k}) - \epsilon a^\dagger(\vec{k})a(-\vec{k}))}.$

## Scalar field – complex

${\cal T}\phi(x){\cal T}^{-1} = \epsilon\phi^{*}(\hat{x})$

## Dirac field

Here:

• ${\cal T}\psi(x){\cal T}^{-1} = T \psi(\hat{x})$
• ${\cal T}\bar{\psi}(x){\cal T}^{-1} = \bar{\psi}(\hat{x}) \gamma^0 T^\dagger \gamma^0,$

where

• $T :\equiv i\gamma^1 \gamma^3 = T^\dagger = T^{-1} = - T^{*},$
• $T \gamma_\mu T^{-1} = \gamma_\mu^T = \gamma^{\mu \ *}.$

Then

• $T u(p,s) = u^*(-p,-s)e^{i\alpha_+(p,s)},$
• $T v(p,s) = v^*(-p,-s)e^{i\alpha_-(p,s)},$

\ni where $\alpha_\pm(p,s) = \pi + \alpha_\pm(-p,-s)$.

Now

• ${\cal U} b(p,s) {\cal U}^{-1} = -b(-p,-s) e^{i\alpha_+(p,s)},$
• ${\cal U} d^\dagger(p,s) {\cal U}^{-1} = -d^\dagger(-p,-s) e^{i\alpha_-(p,s)}.$

Split ${\cal U} = {\cal U}_1{\cal U}_2$ s/t

• ${\cal U}_1 b(p,s) {\cal U}_1^{-1} = e^{i\alpha_+(p,s)} b(p,s),$
• ${\cal U}_1 d^\dagger(p,s) {\cal U}_1^{-1} = e^{i\alpha_-(p,s)} d^\dagger(p,s),$
• ${\cal U}_2 b(p,s) {\cal U}_2^{-1} = -b(-p,-s),$
• ${\cal U}_2 d^\dagger(p,s) {\cal U}_2^{-1} = -d^\dagger(-p,-s).$

Their realizations are:

• ${\cal U}_1 = e^{- i \int d^3\vec{p} \sum_s (\alpha_+(p,s) b^\dagger(p,s)b(p,s) - \alpha_(p,s)d^\dagger(p,s)d(p,s))},$
• ${\cal U}_2 = e^{- i \frac{\pi}2 \int d^3\vec{p} \sum_s (b^\dagger(p,s)b(p,s) + b^\dagger(p,s)b(-p,-s) - d^\dagger(p,s)d(p,s) - d^\dagger(p,s)d(-p,-s))}.$

## Electromagnetic field

• ${\cal T}A^\mu(x){\cal T}^{-1} = A_\mu(\hat{x}),$
• ${\cal U}\vec{a}^a(\vec{k}){\cal U}^{-1} = - (-)^a a^a(-\vec{k}),$
• ${\cal U} = e^{\frac{i\pi}2 \int d^3k (a^\dagger(\vec{k})^a a(\vec{k})_a + (-)^a a^\dagger(\vec{k})^a a(-\vec{k})_a}$.

## 2010.05.30

### String-inspired methods

Filed under: qft — Tags: , , — sandokan65 @ 16:05