TIME (A)SYMMETRY
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The question whether the Nature is time symmetric is fundamental. However, there are different types of symmetries. High energy physics knows ChargeParityTime (CPT) theorem[1], which says that every relativistically invariant dynamics must be symmetric under joint charge conjugation (C), mirror reflection (parity P) and time reversal (T). There are known violations of the individual symmetries (e.g. CP) but never combined CPT. There is also a special preselectionpostselection symmetry [2]. On the other hand, thermodynamics is clearly not time symmetric, because entropy grows with time. The thermodynamic asymmetry does not follow from microscopic evolution but from macroscopic loss of information. Only equilibrium states have constant entropy and time symmetric evolution (if external conditions, e.g. magnetic field, are reversed, too).
Here we consider a different type of symmetry, for measurements. Most measurements, both classical and quantum mechanical, disturb the measured system, they are invasive, so their results will not be time symmetric. However, one can consider a limit of zero strength, when every measurement becomes noninvasive, at the expense of a large noise, due to initial uncertainty of the detector. The total probability of measurement results is the convolution P = D ∗ Q, where D is the internal detector noise and Q(a_{1},...,a_{n}) is the inherent statistical distribution of the system, a function of results of measuring quanities A_{1},...,A_{n} at times t_{1},...,t_{n}, scaled with the measurement strength. The question is now: are the noninvasive measurement results time symmetric under joint time reversal of the state and measured observables? This symmetry is independent of CPT, while preselectionpostselection symmetry is preserved for invasive measurements. Our symmetry has also nothing in common with the second law of thermodynamics (entropy production), because it can be applied to microscopic evolution (when the second law is absent) or equlibrium systems (when the second law predicts symmetry). So this is a new, subtle and fundamental question without any trace in the literature, except our own work.
A measurement is noninvasive if it does not change the state [3]. Mathematically this means that removing the measurement k should not change the results of future measurments
∫ da_{k} Q(a_{1},...,a_{n}) = Q(a_{1},...,/a_{k},...,a_{n})where the slash denotes that the kth measurement is not performed at all. If the property holds all measurements then they all are noninvasive. It may happen that some measurements are noninvasive for each other so the noninvasiveness is stronger if it holds a fixed measurements k but arbirary other measurements. Then the whole state is not disturbed.
For each observable and state there exists time reversal operation ^{T} (e.g. position is unchanged, x^{T} = x, while momentum is reversed p^{T} = −p). If we reverse all observables, times and the initial state ρ → ρ^{T}, then the time symmetry of measurements means
Q(a_{1}(t_{1}),...,a_{n}(t_{n})) = Q^{T}(a_{1}^{T}(−t_{1}),...,a_{n}^{T}(−t_{n}))where we compare normal (Q) and reversed (Q^{T}) states. In such a form, the above condition is independent of CPT symmetry. The symmetry is important e.g. in the principle of the detailed balance and reciprocal relations [4].
Both classical and quantum measurements can be described in an analogous way [5]. The interaction between the detector and the system scales with the strength g. The simplest interaction model assumes instant interation, gpA, where p is detector's momentum and A is the measured quantity. The position q of the detector is shifted by gA, so P(q) = ∫ da D(qga)Q(a). Here Q still depends on g, but has a well defined limit for g → 0:
Q(a)=<δ(a_{n} − A^{c}_{n}(t_{n}))⋅⋅⋅δ(a_{1} − A^{c}_{1}(t_{1}))>,<a_{n}⋅⋅⋅a_{1}>_{Q} = <A^{c}_{n}(t_{n})⋅⋅⋅A^{c}_{1}(t_{1})>
for t_{n} ≥ ... ≥ t_{1}. Here < ⋅ > denotes averages with density ρ: classical phase space integral ∫ dΓ ⋅ ρ and quantum trace Tr ⋅ ρ. The above expression certainly satisfies noninvasiveness. The (super)operation A^{c} acts as A^{c}B =(AB+BA)/2 However, it is commuting operation only classically but not quantum mechanically, because A and B are then operators (matrices), so the order of quantum measurements cannot be reversed. therefore the time symmetry is valid classically but not quantum mechanically (with some exceptions)!!! It it often claimed that quantum mechanics gives nonclassical results. This however, refers to particular classical models, not all. One can always invent a different classical model, explaining a quantum result. Our result cannot be explained by any classical model!
CLASSICAL 
QUANTUM 

NONINVASIVENESS 
YES 
YES 
TIME SYMMETRY 
YES 
NO!!! 
The published paper:
Noninvasiveness and time symmetry of weak measurements
[PDF (IOP Copyright)] [New Journal of Physics 15 023043 (2013)][arXiv:1108.1305]
We, Kurt Franke (formerly at University of Konstanz, now at CERN)
and Wolfgang Belzig (University of Konstanz) and me, have published the above result
in New Journal of Physics. You can find there detailed construction
of the measurements models, quantumclassical analogies and differences, examples
of quantum violation of time symmetry, and further discussion. There is accompanying videoabstract and general scientific summary.
The paper has been featured in Highlights of 2013 of New Journal of Physics, High Energy Particle Physics.
Comment on peerreview criticism
Before publishing in New Journal of Physics, our result underwent long peerreview, with several rejections, including New journal of Physics itself (!). HERE I present excerpts form the reports/editorial decisions, with a comment. If you are interested in full correspondence, it is available on request by email: abednorz[at]fuw.edu.pl.
References
[1] R.F. Streater and A.S. Wightman, PCT, Spin and Statistics and All That ( Benjamin, New York, 1964)
[2] Y. Aharonov Y, P.G. Bergmann and J.L. Lebowitz, Phys. Rev. 134, B1410 (1964);
M. GellMann and J. Hartle, in: Physical Origins of Time Asymmetry, eds. J Halliwell, J PerezMercader and
W. Zurek (Cambridge University Press, 1994) p. 311,
arXiv:grqc/9304023
[3]
L. Onsager, Phys. Rev. 37, 405 (1931);
N.G. van Kampen, Stochastic Processes in Physics and Chemistry (NorthHolland, Amsterdam, 2007)
[4] Y. Aharonov Y, D.Z. Albert and L. Vaidman, Phys. Rev. Lett.
60, 1351 (1988)
[5]
A.J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)