text: 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, capturing the actual reversal of the time coordinate, the T that is an ordinary finite dimensional matrix, acting on spinors and vectors, and the T that is an operator on an infinite-dimensional Hilbert space.
  For a real (not complex) classical (unquantized) scalar field , the time reversal involution can simply be written as
  as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign . A slightly more formal way to write this is
  which has the advantage of emphasizing that  is a map, and thus the "mapsto" notation  whereas  is a factual statement relating the old and new fields to one-another.
  Unlike scalar fields, spinor and vector fields  might have a non-trivial behavior under time reversal.  In this case, one has to write
  where  is just an ordinary matrix. For complex fields, complex conjugation may be required, for which the mapping   can be thought of as a 2x2 matrix. For a Dirac spinor,  cannot be written as a 4x4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8x8 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 ; 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  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
  where  is the y-component of the angular momentum operator and  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  In this case,
  Covariant tensor indexes will transform as  and so on. For quantum fields, there is also a third T, written as  which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields  as
  This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two 's are required.
  All three of these symbols capture the idea of time-reversal; they differ with respect to the specific 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  or  or  depending on context, left for the reader to infer.