What are matrix elements in QFT?

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Discussion Overview

The discussion centers on the concept of matrix elements in quantum field theory (QFT), particularly focusing on the distinction between In and Out states in the context of interactions. Participants explore how these states relate to the calculation of matrix elements using Feynman diagrams and the implications of interactions on these states.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that the matrix element \(\langle p' |\phi^4(x)|p\rangle\) involves an incoming particle with momentum \(p\) and an outgoing particle with momentum \(p'\), and questions whether the bra and ket represent different states due to their temporal labels.
  • Another participant argues that without interaction, the states are the same, implying that the non-interaction scenario leads to the identity operator where nothing changes.
  • A further contribution clarifies that with interaction, the bra and ket are indeed different, as the bra represents an Out-state and the ket an In-state, leading to the conclusion that Feynman diagrams are used to calculate matrix elements between these states.
  • One participant questions whether matrix elements can exist between two In-states and how such calculations would be performed, noting that Feynman diagrams seem applicable only to In and Out states.
  • Another participant counters that even with interaction, the states \(|p\rangle\) and \(\langle p|\) can still be considered the same in the absence of interaction, suggesting that the definitions of In and Out states may not apply in that context.

Areas of Agreement / Disagreement

Participants express differing views on the nature of matrix elements in QFT, particularly regarding the definitions and implications of In and Out states. There is no consensus on whether matrix elements can be defined between two In-states or how to approach such calculations.

Contextual Notes

The discussion highlights the complexity of defining states in QFT, particularly in relation to interactions and the use of Feynman diagrams. Limitations include the potential ambiguity in the definitions of In and Out states and the implications of interactions on these states.

geoduck
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Suppose you want the 1-particle matrix elements of an operator in QFT, e.g.

\langle p' |\phi^4(x)|p\rangle

It seems you would calculate this perturbatively by first Fourier transforming the x-variable to q, assuming an incoming particle with momentum p, an outgoing particle with momentum p', and drawing all interactions vertices but making sure to include one φ4 vertex that also has momentum q entering it.

However, if you do this, aren't |p> and |p'> the Heisenberg states that have momentum p and p' at t=-∞ and t=∞?

Does this means that in \langle p |\phi^4(x)|p\rangle,

<p| and |p> are not the same state? |p> is the Heisenberg state that looks like it has momentum p at t=-∞, while <p| is the Heisenberg state that looks like it has momentum p at t=∞. They have the same label, but they are different states (one is an In state, the other an Out state).

So when one speaks of a matrix element in QFT, does one mean a matrix element whose ket is an In state, and whose bra is an Out state? This seems to be the only type of matrix element that is calculable with Feynman diagrams?
 
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Without interaction, those states are the same. In the absence of interaction, all initial and final states are the same, your non-interaction is the identity operator and nothing happens.
Your interaction then gives transitions between states, but those states are still the same.
 
Without interaction, then in an expression like \langle p |\phi^4(x)|p\rangle, the bra and the ket are the same state: <p|p>=1.

However, with interaction, <p|p>≠1, since the bra is an Out-state, and the ket is an In-state.

It seems if you calculate \langle p |\phi^4(x)|p\rangle with Feynman diagrams, then you are calculating the matrix element between an In-state and an Out-state.

So I was wondering if a matrix element in QFT is defined as being between a ket that is an In-state, and a bra that is an Out-state.

Or can a matrix element be between two In-states for instance? If so, how would you calculate that, as it seems you can't use Feynman diagrams, as Feynman diagrams are for kets that are In-states, and bras that are Out-states.
 
geoduck said:
However, with interaction, <p|p>≠1, since the bra is an Out-state, and the ket is an In-state.
This is still 1 as you do not have an interaction here. The p states are defined as states without interaction, where "in" and "out" are meaningless.
 

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