# A question about time inversion [Weinberg]

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• Antarres
In summary, the discussion is focused on the treatment of degenerate multiplets in Appendix C of Weinberg QFT, Vol 1. The equations in question involve the action of time reversal on a massive one-particle state, with the matrix ##\mathcal{T}_{mn}## being unknown. The speaker is confused about why this matrix must be unitary, as it is derived from an antiunitary operator. They mention that antiunitary operators cannot have a matrix representation, but can be written as a composition of a unitary operator and complex conjugation. The analysis after this point is relatively easy to understand, but the speaker is seeking a more rigorous explanation for the unitarity of ##\mathcal{T}_{mn}##
Antarres
Recently I've started reading Weinberg QFT, Vol 1, to introduce myself better to quantum field theory, after finishing courses in qft at the university. Reading through the second chapter, I've found myself confused by his treatment of degenerate multiplets in Appendix C. Here is the equations that confuses me along with some text around it.
...
To explore the general posibilities of time reversal, let us assume that on massive one-particle state it has the action
$$T\Psi_{\textbf{p},\sigma,n} = (-1)^{j-\sigma}\sum_m\mathcal{T}_{mn}\Psi_{-\textbf{p},-\sigma,m}$$
where p, j, σ are particle's momentum, spin and spin z-component, respectively, and n,m are indices labeling members of a degenerate multiplets of particle species. (The appearance of ##(-1)^{j-\sigma}## and the reversal of ##\textbf{p}## and ##\sigma## are deduced in the same way as in Section 2.6.) The matrix ##\mathcal{T}_{mn}## is unknown, except that since T is antiunitary ##\mathcal{T}## must be unitary.
...

This last statement doesn't seem obvious to me. As far as I know, antiunitary operators in quantum mechanics cannot have a matrix representation(because they are antilinear, unlike matrices), so it is customary to write them in the form ##T = UK##, where ##U## is unitary, and ##K## is complex conjugation. This doesn't help here(at least I don't see it helping), since Weinberg chose to represent time reversal operator by it's action on states, from which he derives the ##(-1)^{j-\sigma}## factor and reversal of momentum and helicity, when acting on the one-particle states of massive particles. (It's derived in section 2.6. The derivation is relatively short.) So I don't see why ##\mathcal{T}## has to be unitary in the previous equation, I couldn't really approach the problem from any side so far, so some help would be appreciated. The analysis that proceeds after this seems relatively easy to understand, I just have problem with this beginning of the analysis.

P.S. I put this article here, judging by the theme of the book. Maybe it could be put in Homework and exercises, too, you're free to move it wherever it's more suitable. I wouldn't consider it homework or anything, since it's just self-learning, so I wanted to clarify this to myself.

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I've come to a conclusion to this problem that is based on my intuition. Basically, if antiunitary operator can be represented as composition of unitary action and conjugation, and if this conjugation aspect is described in a standard way by action on each component of a degenerate multiplet, resulting in the phase that is introduced, than I assume that the mixing of components should have no other option than to be unitary. So for that reason the ##\mathcal{T}## matrix that mixes the components is unitary.

This approach is just my intuition I've developed thinking about the problem, but it doesn't really seem rigorous to me, so I hoped with this question that there is some more direct(rigorous) reason for this conclusion. But maybe there is nothing more than that.

## 1. What is time inversion in physics?

Time inversion, also known as time reversal, is a concept in physics where the direction of time is reversed. This means that the laws of physics would still hold true if time were to move backwards instead of forwards. It is a hypothetical concept that has not been observed in nature.

## 2. How does time inversion differ from time travel?

Time inversion and time travel are often confused, but they are two different concepts. Time inversion is a hypothetical concept in which the direction of time is reversed, while time travel is the ability to physically travel to different points in time. Time inversion does not involve any physical movement, but rather a reversal of time in a specific scenario.

## 3. Can time inversion occur in real life?

As of now, there is no evidence to suggest that time inversion can occur in real life. It is a theoretical concept that is used in certain areas of physics, such as quantum mechanics and thermodynamics, to help understand certain phenomena. However, scientists continue to study and explore the possibility of time inversion in various contexts.

## 4. How is time inversion related to entropy?

Entropy is a measure of disorder or randomness in a system. Time inversion is often used in the context of thermodynamics to explain the concept of entropy. In a time-inverted scenario, the entropy of a system would decrease instead of increase, which is the opposite of what is observed in our universe. This helps scientists better understand the fundamental laws of thermodynamics.

## 5. Is time inversion the same as time dilation?

No, time inversion and time dilation are different concepts. Time dilation, as described by Einstein's theory of relativity, is the slowing down of time for an object in motion relative to an observer. Time inversion, on the other hand, is the reversal of time in a specific scenario. Time dilation is a proven phenomenon, while time inversion is still a theoretical concept.

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