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The simplest example in condensed matter physics that spontaneously breaks time reversal symmetry is a ferromagnet. Because spins (angular momentum) change sign under time reversal, the spontaneous magnetization in the ferromagnet breaks the symmetry. This is a macroscopic example. The chiral spin liquid (Wen-Wilczek-Zee) mentioned in the question is a ...


8

There are numerous research groups engaged in a search for an electric dipole moment of the electron, which, if it exists, would violate time-reversal symmetry. You can see this because any dipole moment the electron might have would need to be parallel to the spin (or anti-parallel). When you reverse time, the spin necessarily flips, but the electric dipole ...


8

The point is that the symmetries in QM (bijective operations sending states to states preserving the transition probability) can be represented by either unitary or antiunitary operators. This is the statement of a famous theorem due to Wigner. It is possible to prove also that, if the Hamiltonian of a system is bounded below, time reversal must be ...


7

As the neutron is not point-like, consider it has a continuous distribution of charge $\rho(\mathbf{r})$ confined in a volume $\Omega$. The dipole electric moment is then given by $\mathbf{D}(\mathbf{r})=\int_\Omega \rho(\mathbf{r}')\delta(\mathbf{r}-\mathbf{r}')d^3r'$ where the coordinates are measured from the centre of mass of the ...


6

Conservation of energy follows from invariance under translation in time, not inversion. This symmetry states that no matter when you do your experiment, it will give the same results. All isolated systems obey this symmetry (and therefore conserve energy) and no violation of it has ever been detected. (Needless to say, it would be a huge event if it were.) ...


5

Reference (page $13$, formula $17.71$) The time-reversal operator is $\Theta = Ke^{-i\pi S_y/\hbar}$, where $K$ is the complex conjugation operator. Taking a spin $1/2$, we have a wavefunction which is a $2$- component spinor $\psi(x) = \begin{pmatrix} \psi_+(x) \\ \psi_-(x) \end{pmatrix}$, Note that, for spin $1/2$, $e^{-i\pi S_y/\hbar} = e^{-i \large ...


5

The microscopic laws of physics are reversible or, to say the least, CPT-symmetric (processes are invariant if they're run backwards in time, in mirror, and with antiparticles). The CPT symmetry follows from the Lorentz symmetry. Langton's ant as well as pretty much any other Turing machine or cellular automaton fails to be microscopically reversible; ...


5

You ask: From my studies in quantum mechanics, I don't remember any postulates stating anything like this, but this all makes sense to me. Are there any theories out there that go along these lines? Indeed there is such a theory. It's called decoherence. You mention the comparison with thermodynamics, and this is basically the same way decoherence ...


5

The functions $-iEt$ and $-i(-E)(-t)$ are exactly the same so they obviously correspond to the same sign of energy if they appear in the exponent defining $|\psi\rangle$. It seems that you think that you may freely replace $t$ by $-t$ and change nothing else. However, this operation isn't a symmetry of the laws of physics, as you have actually demonstrated ...


5

As John Rennie wrote in his answer, what one should consider is not a generic $\Delta t$ but the period $\tau$ which is a positive number. However positivity also arises form coherence with other relations. In particular, in relativistic quantum mechanics $E= \sqrt{\hbar^2 \vec{k}^2 + m^2c^4}$. For $m=0$ you have $E= \hbar |\vec{k}| = \hbar \omega = h\nu$ ...


5

You have all the elements in your question, your difficulty is about what is meant by "time reversal symmetry". Time reversal symmetry holds if, when "playing backwards", the motion observed obeys the same law. With friction it is not the case : friction opposes movement, when playing backwards it (seemingly) promotes it. You can also go to equations for ...


4

The English Wikipedia article on magnetic monopoles has the following equation for the 'extended' Lorentz-Force of a magnetic field on a electrically and magnetically charged particle: $$ \vec{F}=q_{\mathrm e}\left(\vec{E}+\vec{v}\times\vec{B}\right) + q_{\mathrm m}\left(\vec{B}-\vec{v}\times\frac{\vec{E}}{c^2}\right) $$ Under time reversal ($t$ is ...


4

At research level, you might be interested in the PDG review on conservation laws: http://pdg.lbl.gov/2009/reviews/rpp2009-rev-conservation-laws.pdf Also, the review about CPT invariance gives information about tests of CPT violation in neutral kaons, but I can apparently only post one hyperlink (reputation too low). Note that CP violation itself is still ...


4

Just a few pointers for you to explore more on this. Check out Aharonov's paper the time symmetric formulation of quantum mechanics: http://arxiv.org/abs/quant-ph/9501011 Tony Leggett talks about this: http://www.youtube.com/watch?v=IGim9uzcumk It's a nice video and quite simple to understand.


4

Energy, frequency and period are all scalars. It's correct to write: $$ E = \frac{h}{\tau} $$ where $\tau$ is the period, but the sign of $\tau$ does not change if you run time backwards so the sign of energy doesn't change in QM either.


3

Maybe the article was about CP violation (http://en.wikipedia.org/wiki/CP_violation ), which seems to imply violation of the symmetry with respect to time reversal.


3

$PT-, T-, P-$ transformations refer to subgroup of discrete transformations of the Lorentz group. They transform connected components of the Lorentz group between each other ($PT$ transformation transforms $L^{\uparrow}_{+}$ representation to $L^{\downarrow}_{+}$). In general, they can't be represented as the special case of rotation, which refer to subgroup ...


3

The the easiest way to see that time reversal transforms electrons into positrons relies on the fact that PCT (parity, charge conjugation and time reversal) combined are a symmetry of every Lorentz-invant QFT. Using $P^{-1} = P$, $C^{-1} = C$, $T^{-1} = T$, i.e. a parity transformation is undone by a second parity transformation etc. you can see that $$1 = ...


3

The sentence in Peskin's and Schroeder's book that "the weak interactions preserve CP and T" is a bit misleading but there is a sense in which it is right. Experimentally, CP and T is known to be violated and CPT is always a symmetry. Theoretically, CPT is always a symmetry, too – it's proven by the CPT theorem. The CPT transformation is effectively a ...


3

It is very important to distinguish whether the symmetry is broken explicitly or spontaneously. I think that the sentence "Now when I break this symmetry spontaneously (or explicitly)" indicates that its author isn't quite distinguishing these things. An explicit symmetry breaking generally lifts the degeneracy because the different parts of the multiplets ...


3

I can not answer you question mathematically, but my experience with Burgers equation tells me that there is no such transformation. If you think of Euler equation as Navier-Stokes in the limit where the viscosity vanishes $\nu \to 0$, then time reversal symmetry is simply spontaneously broken. As long as you have a finite viscosity the system is ...


2

When you perform a time reversal, you have to not only change the sign of $t$ everywhere, but also change the sign of some of the other physical quantities involved. Time reversal symmetry just means that the form of all equations involved is the same after making the appropriate transformations. If two quantities are related by a time derivative, they ...


2

No, the sign of the gap does not represent particle-hole symmetry. The superconducting gap simply being non-zero automatically encodes the existence of particle-hole symmetry. Variations in the sign of the gap in the Brillouin zone (BZ) determines whether a superconductor is topologically trivial or non-trivial. However, topological or not, a superconductor ...


2

Let us here just consider the classical case ($\hbar=0$). I) Noether's theorem does not work for discrete symmetries like time reversal symmetry, $$\tag{1} T: t ~\longrightarrow~ -t, $$ cf. e.g. this Phys.SE post. II) Instead, energy conservation follows from time translation symmetry $$\tag{2} t ~\longrightarrow~ t + a, \qquad a~\in~ \mathbb{R}, $$ ...


2

I) First of all, one should never use the Dirac bra-ket notation (in its ultimate version where an operator acts to the right on kets and to the left on bras) to consider the definition of adjointness, since the notation was designed to make the adjointness property look like a mathematical triviality, which it is not. See also this Phys.SE post. II) OP's ...


2

I don't know the article you refer to, but I believe the Hamiltonian you discuss should get a $\pi$-phase shift after one turn around a (2D) lattice cell. So I guess it should read $H=F^{\dagger}\cdot H_{\pi}\cdot F$ with $$H_{\pi}=t\left(\begin{array}{cccc} 0 & e^{\mathbf{i}\pi/4} & 0 & e^{-\mathbf{i}\pi/4}\\ e^{-\mathbf{i}\pi/4} & 0 & ...


2

Again, thanks to the $SU(2)$ PSG proposed by prof.Wen, I can answer my question now, $THT^{-1}$ is in fact $SU(2)$ gauge equivalent to $H$, and the statement "$H$ is also not SU(2) gauge equivalent to the time-reversal transformed Hamiltonian $THT^{-1}$" in my question is wrong. Let's rewrite the Hamiltonian as ...


2

I think the answer should be 'no'. Because when we introduce the antiunitary time-reversal(TR) opeartor $T$ for spin-system, it should satisfy $T\mathbf{S}_iT^{-1}=-\mathbf{S}_i$ since angular-momentum should be sign reversed under TR(due to the classical correspondence). Thus, spin-spin interactions like $\mathbf{S}_i\cdot\mathbf{S}_j$ are invariant under ...


2

The laws of electromagnetism are indeed time-reversal invariant, but this can sometimes be in a fairly restricted sense, particularly when dealing with causality as regards the Liénard-Wiechert potentials. Consider the following situation. A charged particle is travelling under the influence of some force $\mathbf F_\text{ext}(t)$ which is due to a ...


2

There are two possible answers to why $T^2=-1$: a) Why not. The total phase of a quantum state is unphysical. So a symmetry may be realized as a projective representation. Here T may be viewed as a projective representation of time reversal $T_{phy}$ which satisfy $T^2_{phy}=1$. b) If we define the time reversal symmetry to be realized as a regular ...



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