Why is there an arrow of time?

[DrChinese]

I have a question, hope maybe someone has seen some papers on the subject and can point me in a direction. I searched arxiv/astro-ph with the search paprameter "arrow of time" but could not find anything in the following specific area:

Hypotheses:

a) Time is symmetric.
b) When the big bang occurred, half the mass went forward in time, half went backwards in time.

Anti-matter universe <------- Big Bang --------> Our universe

Would provide a simple explanation of why we see effects dominating in one time direction and not the other... it is a result of initial conditions. I am pretty certain someone must have speculated on this before, there are probably plenty of citations but I can't find the right phrase. Any suggestions? There have been new interpretations of QM which involve time symmetry (I searched on that too) which I suspect would have some cosmological implications. So that is what is driving the question.

 

[Dmitry67]

DrChinese, are you talking about the macroscopic arrow of time or QM arrow of time?

Macroscopic arrow of time I guess had been explained a long time ago as a consequence of the Big Bang (low entropy initial conditions -> Loschmidt's paradox)

QM arrow of time (CP violation = T-symettry violation) is a mystery.

 

[Carid]

Ilya Prigogine wrote a book on this called "The End of Certainty"

http://www.absoluteastronomy.com/topics/Ilya_Prigogine"

 

[BillSaltLake]

There's a "coincidence" that suggests something like the idea of antimatter going backward in time and matter going forward, but this would have occurred much later than the Bang, at around a microsecond. At the time of stable baryon pair production, there were around 10¹⁸ total baryons + photons per Planck mass-equivalent of mass+energy. Because of charge conservation, the number of protons produced must have been exactly equal to the number of antiprotons, but baryon number conservation did not necessarily apply to neutrons and antineutrons.

If (for some unknown reason) there is a conservation principle in which any sample with mass+energy equal to the Planck mass must have its mass+energy conserved despite gaussian statistical fluctuations, then there could be an interesting consequence. This "conservation principle" is the same as saying that there's a node in the gaussian count fluctuations for ensembles equaling the Planck mass.

One reason that mass+energy fluctuations occurred during pair production because the mass per massive particle pair was not changing while the energy per photon pair was decreasing. Converting back and forth several times between photon pairs and massive pairs introduces some variation. Some samples would be expected to end up with less mass+energy than average and some with more. For a sample of 10¹⁸, the standard deviation would be about 10⁹ particles.

Suppose that each Planck-massed sample was brought up to or down to average by spontaneous creation/destruction of neutrons or antineutrons. If the neutrons (in the Feynman-diagram sense) go "forward" in time while the antineutrons go "backward", then a sample that lost excessive mass+energy during the baryon pair production era would supplement that lost mass+energy by creation of neutrons, while a sample that ended up with an excess of mass+energy would spontaneously annihilate some of the antineutrons. The result would be an excess of neutrons with about 1 neutron per 10⁹ photons, which is about the observed baryon:photon ratio.

 

[dleiter]

THE ANSWER TO THE QUESTION: WHY IS THERE AN “ARROW OF TIME”?

In the following two papers has been shown that description of the quantum measurement process can be “completed” by incorporating an operator symmetry of microscopic observer-participation called “Measurement Color” into the formalism of quantum electrodynamics (Leiter, D. 2009), http://journalofcosmology.com/LeiterArrowofTime.pdf
Leiter, D. 2010), http://journalofcosmology.com/LeiterQUANTUMREALITY.pdf)

It has been shown that the Measurement Color operator symmetry within MC-QED generates a spontaneous CPT symmetry breaking effect which causes the photon to carry the quantum electrodynamic arrow of time by virtue of the fact that it has a negative parity under Wigner time reversal. This resolves the age old puzzle about the origin of the arrow of time in the universe.

However even though this result is interesting it is natural to ask if there exists any experimental evidence which supports the idea that “the photon carries the arrow of time”?

The answer is that in new paper to be published in the near future (D. Leiter 2010 in press) it has been shown that recently performed Michelson interferometer experiments, using combinations of ordinary mirrors and phase conjugate mirrors, contain experimental results which offer strong evidence that the photon has a negative parity under Wigner time reversal in support of the MC-QED prediction that “the photon carries the arrow of time”.

 

[Dmitry67]

Whats so special about the photons?
Photon is just one of 16 SM particles.
Whats about W, Z, g?

 

[Orion1]

The law of entropy and the second law of thermodynamics...

In physical cosmology, assuming that nature is described by a Grand unification theory, the grand unification epoch was the period in the evolution of the early universe following the Planck epoch, starting at about 10^{-43} seconds after the Big Bang, in which the temperature of the universe was comparable to the characteristic temperatures of grand unified theories. If the grand unification energy is taken to be 10^{15} GeV, this corresponds to temperatures higher than 10^{27} K. During this period, three of the four fundamental interactions—electromagnetism, the strong interaction, and the weak interaction—were unified as the electronuclear force. Gravity had separated from the electronuclear force at the end of the Planck era.

During the Grand Unification Epoch, physical characteristics such as mass, charge, flavour and colour charge were meaningless. The grand unification epoch ended at approximately 10^{-36}seconds after the Big Bang. At this point several key events took place. The strong force separated from the other fundamental forces. The temperature fell below the threshold at which X and Y bosons could be created, and the remaining X and Y bosons decayed.[citation needed] It is possible that some part of this decay process violated the conservation of baryon number and gave rise to a small excess of matter over antimatter (see baryogenesis). This phase transition is also thought to have triggered the process of cosmic inflation that dominated the development of the universe during the following inflationary epoch.

Baryon asymmetry parameter

The challenges to the physics theories are then to explain how to produce this preference of matter over antimatter, and also the magnitude of this asymmetry. An important quantifier is the asymmetry parameter,

$$\eta = \frac{n_B - n_{\bar B}}{n_\gamma}.$$

This quantity relates the overall number density difference between baryons and antibaryons ($$n_B$$ and $$n_{\bar B}$$, respectively) and the number density of cosmic background radiation photons $$n_{\gamma}$$.

According to the Big Bang model, matter decoupled from the cosmic background radiation (CBR) at a temperature of roughly 3,000 kelvin, corresponding to an average kinetic energy of 3,000 K / (10.08×10^4 K/eV) = $$0.3 \; \text{eV}$$. After the decoupling, the total number of CBR photons remains constant. Therefore due to space-time expansion, the photon density decreases. The photon density at equilibrium temperature $$T$$, per cubic kelvin and per cubic centimeter, is given by

$$n_{\gamma} = \frac{1}{\pi^2} {\left(\frac{k_B T}{\hbar c}\right)}^3 \int_0^\infty \frac{x^2}{\exp(x) - 1} dx \simeq 20.3 \left(\frac{T}{1\text{K}}\right)^3 \text{cm}^{-3}$$

with $$k_B$$ as the Boltzmann constant, $$\hbar$$ as the Planck constant divided by $$2 \pi$$ and $$c$$ as the speed of light in vacuum. At the current CBR photon temperature of $$2.725 \; \text{K}$$, this corresponds to a photon density $$n_{\gamma}$$ of around $$4^{11}$$ CBR photons per cubic centimeter.

Therefore, the asymmetry parameter $$\eta$$, as defined above, is not the "good" parameter. Instead, the preferred asymmetry parameter uses the entropy density $$s$$,

$$\eta_s = \frac{n_B - n_{\bar B}}{s}$$

because the entropy density of the universe remained reasonably constant throughout most of its evolution. The entropy density is

$$s \ \stackrel{\mathrm{def}}{=}\ \frac{\mathrm{entropy}}{\mathrm{volume}} = \frac{p + \rho}{T} = \frac{2\pi^2}{45}g_{*}(T) T^3$$

with $$p$$ and $$\rho$$ as the pressure and density from the energy density tensor $$T_{\mu \nu}$$ and $$g^{*}$$ as the effective number of degrees of freedom for "massless" particles (if $$mc^2 \ll k_B T$$ holds) at temperature $$T$$,

$$g_{*} (T) = \sum_\mathrm{i=bosons} g_i{\left(\frac{T_i}{T}\right)}^3 + \frac{7}{8}\sum_\mathrm{j=fermions} g_j{\left(\frac{T_j}{T}\right)}^3$$

for bosons and fermions with $$g_i$$ and $$g_j$$ degrees of freedom at temperatures $$T_i$$ and $$T_j$$ respectively. At the present era, $$s = 7.04 \cdot n_{\gamma}$$.

The second law of thermodynamics is an expression of the universal principle of decay observable in nature. It is measured and expressed in terms of a property called entropy, stating that the entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium; and that the entropy change dS of a system undergoing any infinitesimal reversible process is given by δq / T, where δq is the heat supplied to the system and T is the absolute temperature of the system.

Entropy (arrow of time)

Entropy is the only quantity in the physical sciences that "picks" a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system will increase when no extra energy is consumed. Hence, from one perspective, entropy measurement can be thought of as a kind of clock—although not really an accurate measure of time.

Physical processes at the microscopic level are believed to be either entirely or mostly time symmetric, meaning that the theoretical statements that describe them remain true if the direction of time is reversed; yet when we describe things at the macroscopic level it often appears that this is not the case: there is an obvious direction (or flow) of time. An arrow of time is anything that exhibits such time-asymmetry.

The largest contributor to the entropy budget of the Universe is super massive black holes.

Schwarzschild radius:
$$R_s = \frac{2GM}{c^2}$$

Schwarzschild black hole sphere surface area:
$$A_s = 4 \pi R_s^2 = 4 \pi \left( \frac{2GM}{c^2} \right)^2 = \frac{16 \pi G^2 M^2}{c^4}$$

Planck radius:
$$r_P = \sqrt{\frac{\hbar G}{c^3}}}$$

Schwarzschild black hole entropy integration by substitution:
$$S_{BH} = \frac{dE}{dT} = k_B \ln \Omega_s = k_B \left( \frac{A_s}{4 r_P^2} \right) = \frac{k_B}{4} \left( \frac{16 \pi G^2 M^2}{c^4} \right) \left( \frac{c^3}{\hbar G} \right) = \frac{4 \pi k_B G}{\hbar c} M^2$$

Schwarzschild black hole entropy:
$$\boxed{S_{BH} = \frac{4 \pi k_B G}{\hbar c} M^2}$$

Entropy of the Universe:
$$S_u = \sum_{i = 1}^{n} S_i = k_B \ln \Omega_u$$

$$S_u = 3.1 \cdot 10^{104} \; \frac{\text{j}}{\text{K}}$$

$$\boxed{\ln \Omega_u = \frac{S_u}{k_B} = 2.245 \cdot 10^{127}}$$
$$\Omega_u$$ - number of equally probable states in the Universe

The law of entropy and the second law of thermodynamics are more fundamental laws in the Universe than those involved with symmetry breaking events such as the phase transition of Planck epoch energy to baryonic matter, grand unification theory, matter over antimatter asymmetry and the Standard Model, etc., because those two laws were in effect prior to these epochs in the Universe during the Planck epoch, when the dimensions of space-time and energy and possible equally probable states were generated.

Reference:
Second law_of_thermodynamics - Wikipedia
http://en.wikipedia.org/wiki/Boltzmann%27s_constant"
http://en.wikipedia.org/wiki/Entropy_%28arrow_of_time%29"
http://en.wikipedia.org/wiki/Arrow_of_time"
http://en.wikipedia.org/wiki/Grand_unification_epoch"
http://en.wikipedia.org/wiki/Baryogenesis#Baryon_asymmetry_parameter"
http://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropy"
http://theastronomist.fieldofscience.com/2009/09/entropy-of-universe.html"
http://en.wikipedia.org/wiki/Schwarzschild_radius#Formula_for_the_Schwarzschild_radius"
http://en.wikipedia.org/wiki/Planck_length#Value"

 

[dleiter]

Hi Dmitry67,

In reply to my comments you asked:

Whats so special about the photons?
Photon is just one of 16 SM particles.
Whats about W, Z, g?

This is a very good question and here is the answer.

Measurement Color Electrodynamics (MC-QED) is a gauge invariant quantum field theory of electrons, positrons, and photons in which a quantum electrodynamic arrow of time emerges due to spontaneous CPT symmetry breaking.

However since it is a gauge field theory Measurement Color generalizations of the quantum electrodynamic formalism to the case of SU(3)xSU(2)xU(1) quantum gauge particle field theories with appropriately chosen Higgs sectors can be formulated. In this higher internal symmetry context the effects of spontaneous CPT symmetry breaking would then imply that the massless photon as well as all of the masssive gauge bosons carry the quantum field theoretic arrow of time.

More comments would be appreciated.

Darryl Leiter

 

[Dmitry67]

I was blocked at the sentence

We begin our discussion by noting that the origin of this problem lies within the nature of Copenhagen Interpretation itself.

As I see, measurement is analyzed from CI perspective, while Copenhagen does not make any sense in new millenium. Whats about the decoherence?

 

[Orion1]

The particle physics weak arrow of time...

The particle physics (weak) arrow of time

Certain subatomic interactions involving the weak nuclear force violate the conservation of both parity and charge conjugation, but only very rarely. An example is the kaon decay [1]. According to the CPT Theorem, this means they should also be time irreversible, and so establish an arrow of time. Such processes should be responsible for matter creation in the early universe.

This arrow is not linked to any other arrow by any proposed mechanism, and if it would have pointed to the opposite time direction, the only difference would have been that our universe would be made of anti-matter rather than from matter. More accurately, the definitions of matter and anti-matter would just be reversed.

That the combination of parity and charge conjugation is broken so rarely means that this arrow only "barely" points in one direction, setting it apart from the other arrows whose direction is much more obvious.

Reference:
http://en.wikipedia.org/wiki/Arrow_of_time#The_particle_physics_.28weak.29_arrow_of_time"
http://en.wikipedia.org/wiki/Irreversible_process_%28thermodynamics%29"

 

[Dmitry67]

For me it is a biggest mystery
I don't know why it is not discussed on these forums.
That 'weak' T-symmetry violation is THE ONLY real T-symmetry violation, because all other 'arrows' (quantum, thermodynamical) are explaned based on the low entropy state at BB

 

[dleiter]

I As I see it, measurement is analyzed from CI perspective, while Copenhagen does not make any sense in new millenium. Whats about the decoherence?

ON SPONTANEOUS CPT BREAKING AND DECOHERENCE IN MC-QED

First of all if you look more carefully at the two papers on MC-QED and the arrow of time

http://journalofcosmology.com/LeiterArrowofTime.pdf

http://journalofcosmology.com/LeiterQUANTUMREALITY.pdf

you will see that MC-QED does not use the CI since it contains its own dynamically determined interpretation.

Second, by virtue of the Measurement Color symmetry within MC-QED, operators associated with localized fermionic “system + apparatus” can be defined in terms of a finite range of fermion Measurement Colors. In this context a reduced density matrix operator, associated with the localized fermionic (system + apparatus) in the presence of an external environment, can be defined by tracing over the states of the “environment”.

Then because of the spontaneous CPT symmetry breaking in the MC-QED formalism, the expectation value of the reduced Von Neumann entropy operator for the (system + apparatus) in the presence of an external environment has a time dependence which is both non-unitary and time-reversal violating.

Hence MC-QED contains a combination of (decoherence + dissipation) which occurs in the context of the expectation value of the reduced “Von Neumann Entropy operator, that describes a finite (system + apparatus) in the presence of an external environment by tracing over the states of the “environment”.

 

[Orion1]

The quantum arrow of time

According to the Copenhagen interpretation of quantum mechanics, quantum evolution is governed by the Schrödinger equation, which is time-symmetric, and by wave function collapse, which is time irreversible.

Unsolved problems in physics: What links quantum arrow of time to the thermodynamic arrow?

In my opinion, the quantum arrow of time and the thermodynamic arrow of time are linked by entropy. Time irreversibility asymmetry and symmetry breaking events and the Schrödinger equation, which is time-symmetric, and wave function collapse, which is time irreversible, all obey the law of entropy and the second law of thermodynamics.

Planck energy:
$$E_P = \sqrt{\frac{\hbar c^5}{G}}$$

Planck temperature:
$$T_P = \frac{m_P c^2}{k_B} = \sqrt{\frac{\hbar c^5}{G k_B^2}}$$

Planck entropy integration by substituion:
$$S_P = \frac{dE}{dT} = \frac{E_P}{T_P} = \left( \sqrt{\frac{\hbar c^5}{G}} \right) \left( \sqrt{\frac{G k_B^2}{\hbar c^5}} \right) = k_B$$

$$\boxed{\ln \Omega_P = 1}$$

Planck entropy:
$$\boxed{S_P = k_B \; \; \; \Omega_P = e}$$

Number of equally probable Planck states:
$$\boxed{\Omega_P = e = 2.71828}$$

Reference:
http://en.wikipedia.org/wiki/Arrow_of_time#The_weak_arrow_of_time"
http://en.wikipedia.org/wiki/Boltzmann%27s_constant"
http://en.wikipedia.org/wiki/Planck_energy"
http://en.wikipedia.org/wiki/Planck_temperature"
http://en.wikipedia.org/wiki/Planck_entropy"
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29"

 

[dleiter]

Reply to Orion1 as to "Why is there an arrow of time"

Orion1 commented that:

In my opinion, the quantum arrow of time and the thermodynamic arrow of time are linked by entropy. Time irreversibility asymmetry and symmetry breaking events and the Schrödinger equation, which is time-symmetric, and wave function collapse, which is time irreversible, all obey the law of entropy and the second law of thermodynamics.

In reply dleiter wishes to point out that in the context of the papers on Measurement Color Quantum Electrodynamics (MC-QED)
http://journalofcosmology.com/LeiterArrowofTime.pdf
http://journalofcosmology.com/LeiterQUANTUMREALITY.pdf
the reverse of the logic in Orion1's comment occurs.

This is because by virtue of the Measurement Color symmetry within MC-QED, operators associated with localized fermionic “system + apparatus” can be defined in terms of a finite range of fermion Measurement Colors. Then a reduced form of the “Von Neumann Entropy operator”, which is associated with the localized fermionic (system + apparatus) in the presence of an external environment, can be defined by tracing over the states of the “environment”.

Because of the spontaneous CPT symmetry breaking in the MC-QED formalism, the expectation value of the reduced Von Neumann entropy operator for the (system + apparatus) in the presence of an external environment in has a time dependence which is both non-unitary and time-reversal violating.

Hence in the context of the expectation value of the reduced “Von Neumann Entropy operator which describes a finite (system + apparatus) in the presence of an external environment by tracing over the states of the “environment”, a quantum electrodynamic description of the Second Law of Thermodynamics can be obtained by generalizing on the work done in section 4.1 of the book by Zeh 2007 and section 2.4.3 of the book by Schlosshauer 2007).

In MC-QED this is due to the effects of Measurement Color which creates the spontaneous CPT symmetry breaking in the formalism and causes the photon operator to carry the arrow of time, also underlies the origin of the Second Law of Thermodynamics in the quantum electrodynamic context.

Since result is not limited to quantum electrodynamics alone it can be shown that Measurement Color generalizations of SU(3)xSU(2)xU(1) quantum gauge particle field theories (associated with the Standard Model and Grand Unified Models) can be constructed in the context of which the effects of spontaneous CPT symmetry breaking cause the photon as well as the massive gauge bosons carry the quantum field theoretic arrow of time.

Hence in this manner the idea that the origin of the Second Law of Thermodynamics can be connected to spontaneous CPT symmetry breaking effects in the universe can be extended to include all elementary particle interactions.

 

[Orion1]

Riemann zeta function...


CBR photon equilibrium temperature:
$$T_{\gamma} = 2.725 \; \text{K}$$

CBR photon density at equilibrium temperature $$T_{\gamma}$$:
$$n_{\gamma} = \frac{1}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3 \int_0^\infty \frac{x^2}{e^x - 1} dx$$

This is my solution for the integration equation on post #7.

Integration:
$$\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \sum_{n=1}^\infty n^{-3} = 2 \zeta(3)$$

Complex variable:
$$\boxed{s = 3}$$

$$\boxed{\int_0^\infty \frac{x^2}{e^x - 1} dx = 2 \zeta(3)}$$

Where $$\zeta(s)$$ is the Riemann zeta function.

Riemann zeta function:
$$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$

The CBR photon density at equilibrium temperature $$T_{\gamma}$$:
$$\boxed{n_{\gamma} = \frac{2 \zeta(3)}{\pi^2} {\left(\frac{k_B T_{\gamma}}{\hbar c}\right)}^3}$$

$$\boxed{n_{\gamma} = 4.105 \cdot 10^{8} \; \frac{\text{photons}}{\text{m}^3}}$$

Are these equations correct?

Reference:
http://en.wikipedia.org/wiki/Riemann_zeta_function"
http://en.wikipedia.org/wiki/Baryogenesis#Baryon_asymmetry_parameter"
http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation"

 

[cristo]

The "journal of cosmology" is not a reputable journal, and therefore cannot be used as a reference. Since the OP has not returned for several months, this thread is closed. If DrChinese wishes to revisit this question, he should feel free to open a new thread.