In fact, this in reference to parity-time symmetry of a system. We can have systems having PT-symmetry, but what exactly does it mean when it comes to time-reversal symmetry?
It means that if you "halt" a system at some time and then let it "go backwards" (i.e. reverse *all* velocities by their negative values) then the system will evolve exactly on the trajectory it had come to the point of reversal -- in reverse order of course.
One well-known means to break time reversal symmetry is to have a finite applied magnetic field.
In quantum mechanics, as a result of time reversal symmetry, each single electron state must be at least two-fold degenerate (so-called Kramers doublets) [edit: in the absence of a magnetic field]. Once you apply the magnetic field this degeneracy is typically lifted.
I think there is also something related with the Evolution Operator. If the system admits time reversal then the norm of this operator should be the same. In this notes
http://math.mit.edu/~stevenj/18.369/wave-equations.pdf
the author discuss about it for wave-like systems.
In quantum mechanics, time reversal symmetry implies that under t going to -t and wavefunction going to its complex conjugate, the Schrodinger equation must remain form invariant.
In quantum field theory, this translates as invariance under similarity transformation of operators caused by an anti-unitary operator.
In classical theory, time reversal is much simpler. All one needs to ensure is that the dynamical equations remain form invariant under
t going to -t and all the momenta going to -momenta.
Samit, anti-linear operators are those such that A(c1 u + c2 v) = c1* A u + c2* A v, where * is complex conjugation. Since in quantum mechanics, under a time reversal, psi goes to psi*, the time reversal operator necessarily has to be anti-linear. And since symmetry operators are unitary, in the case of time reversal symmetry, this corresponds to anti-unitary operator. See Wigner's work.
Maybe one should remark, that all these approaches depend on the chosen representation (the "model") and it's intrinsic Interpretation of time with respect to one-parameter transformations of the respective systems... they most often deviate from the definition of "time" within a local coordinate representation of the system where the measurement happens...
As long as we talk on physics, the only measurement you can perform is a measurement in a setup where you have identified the relevant objects, the (physical) transformations and your (local) space-time coordinates exactly. All this Information is a prerequisite. However, that's messed up in a lot of cases by just making implicit assumptions or forgetting to specify one or more of the above requirements. There are simple examples where you find one and the same (algebraic) equation holding valid for *different* objects, however, the standard Interpretation of such equations with respect to spacetime description is that people implicitly talk about points (or even special coordinates - mostly euclidean affine...). On the other hand, you can find various, mathematically equivalent representations for one and the same object (e.g. a (real) vector represented by products of conjugated spinors, etc.) which now of course allows for formally *different* transformations to be discussed with respect to that object. So if you transform the object representations by groups, you will obtain apparently different "behaviour". That's where (due to the group transformations) a parameter enters which translates the system/the objects. People often do not differentiate such Parameters from "time" parametrization.
As such, you will find in literature a lot of discussions where people make statements about "time", however, the (physical) notion time is fixed to your local coordinate system. So in order to handle this properly one has to transform the respective system and introduce projections onto local coordinates in order to tell something about time. If you look into most of the models, the projection is replaced by assumptions and suggestions and not by mathematics (e.g., in QFT, the standard Approach to handle this is to introduce something like "plane waves" and Green's functions and some additional assumptions and "explanations" to calm people...). And moreover, most of the publications forget that each algebraic manipulation corresponds to changes in the geometrical/observable setup of the physical system and its description. A simple example is naive complexification of coordinates.
P.S. With respect to time-reversal, the standard argument for such a discussion is usually that an apparent (formal) algebraic symmetry "permits" this reflection due to some (mostly formal) invariance under t -> -t. However, such formal arguments are related to the respective representation only, and not necessarily related to physics and physical observations. A better or another choice of representing the objects will eventually NOT lead to such symmetries... There are similar issues in spinor reps when people start to discuss significance of signs in Riemannian spaces although there is an intrinsic identification of certain ("point") representations... So after all, my only message is to think carefully about what you are using and doing ;-))) this should contradict in some places some of the standard textbooks or publications on these topics ;-)))
@Samit
My view on TRS is the following
(1) Newtons law remains valid . For example in SHO only x=A cos wt is only allowed.
(2) Quantum mechaniçally one can generate many Hamiltonians which are not PT symmetry but sayisfy TRS. They have real spectra. However it is not easy to realise the
same . However if you use a stardard procedure , then it is very easy.
Prof B.Rath
In order to move in low-Re, or Stokes flow, the driving deformation of the device must not be time-symmetric. How can I understand the time-symmetric in this method?
Dear Prof. Samit Kumar Gupta
In exploration geophysics (mostly for oil & gas) there is a numerical technique, where time inversion for partial differential equations is very useful: they call it "time-reversal (TR) focusing" where the waveforms received at transducers are flipped in time & sent back resulting in a wave converging at the original source ( it happen regardless of the complexity of the propagation medium, that is, heterogeneity and anisotropy). Of course, there are simplifications such as only consider acoustic wave propagation or the medium to be homogeneous.
This TR study can be done using the elastic (2/3) or acoustic (1) wave equations.
Article Time-reversal in geophysics: the key for imaging a seismic s...