Editing T-symmetry (section)

===Formal notation===
In formal mathematical presentations of T-symmetry, three different kinds of notation for '''T''' need to be carefully distinguished: the '''T''' that is an [[involution (mathematics)|involution]], capturing the actual reversal of the time coordinate, the '''T''' that is an ordinary finite dimensional matrix, acting on [[spinor]]s and vectors, and the '''T''' that is an operator on an infinite-dimensional [[Hilbert space]].

For a [[real number|real]] (not [[complex number|complex]]) classical (unquantized) [[scalar field]] <math>\phi</math>, the time reversal [[involution (mathematics)|involution]] can simply be written as

:<math>\mathsf{T} \phi(t,\vec{x}) = \phi^\prime(-t,\vec{x}) = s\phi(t,\vec{x})</math>

as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign <math>s=\pm 1</math>. A slightly more formal way to write this is

:<math>\mathsf{T}: \phi(t,\vec{x}) \mapsto \phi^\prime(-t,\vec{x})  = s\phi(t,\vec{x})</math>

which has the advantage of emphasizing that <math>\mathsf{T}</math> is a [[map (mathematics)|map]], and thus the "mapsto" notation <math>\mapsto ~,</math> whereas <math>\phi^\prime(-t,\vec{x})  = s\phi(t,\vec{x})</math> is a factual statement relating the old and new fields to one-another.

Unlike scalar fields, [[spinor]] and [[vector field]]s <math>\psi</math> might have a non-trivial behavior under time reversal.  In this case, one has to write

:<math>\mathsf{T}: \psi(t,\vec{x}) \mapsto \psi^\prime(-t,\vec{x}) = T\psi(t,\vec{x})</math>

where <math>T</math> is just an ordinary [[matrix (mathematics)|matrix]]. For [[complex number|complex]] fields, [[complex conjugation]] may be required, for which the mapping  <math>K: (x+iy) \mapsto (x-iy)</math> can be thought of as a 2×2 matrix. For a [[Dirac spinor]], <math>T</math> cannot be written as a 4×4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8×8 matrix, acting on the 8 real components of a Dirac spinor.

In the general setting, there is no ''ab initio'' value to be given for <math>T</math>; its actual form depends on the specific equation or equations which are being examined. In general, one simply states that the equations must be time-reversal invariant, and then solves for the explicit value of <math>T</math> that achieves this goal.  In some cases, generic arguments can be made. Thus, for example, for spinors in three-dimensional [[Euclidean space]], or four-dimensional [[Minkowski space]], an explicit transformation can be given. It is conventionally given as

:<math>T=e^{i\pi J_y}K</math>

where <math>J_y</math> is the y-component of the [[angular momentum operator]] and <math>K</math> is complex conjugation, as before. This form follows whenever the spinor can be described with a linear [[differential equation]] that is first-order in the time derivative, which is generally the case in order for something to be validly called "a spinor".

The formal notation now makes it clear how to extend time-reversal to an arbitrary [[tensor field]] <math>\psi_{abc\cdots}</math> In this case,

:<math>\mathsf{T}: \psi_{abc\cdots}(t,\vec{x}) \mapsto \psi_{abc\cdots}^\prime(-t,\vec{x})  = {T_a}^d \,{T_b}^e \,{T_c}^f \cdots \psi_{def\cdots}(t,\vec{x})</math>

Covariant tensor indexes will transform as <math>{T_a}^b = {(T^{-1})_b}^a</math> and so on. For quantum fields, there is also a third '''T''', written as <math>\mathcal{T},</math> which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields <math>\Psi</math> as

:<math>\mathsf{T}: \Psi(t,\vec{x}) \mapsto \Psi^\prime(-t,\vec{x})  = \mathcal{T} \Psi(t,\vec{x}) \mathcal{T}^{-1}</math>

This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two <math>\mathcal{T}</math>'s are required.

All three of these symbols capture the idea of time-reversal; they differ with respect to the specific [[space (mathematics)|space]] that is being acted on: functions, vectors/spinors, or infinite-dimensional operators.  The remainder of this article is not cautious to distinguish these three; the ''T'' that appears below is meant to be either <math>\mathsf{T}</math> or <math>T</math> or <math>\mathcal{T},</math> depending on context, left for the reader to infer.