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This question is about expressions of the form $$ \langle x_f, t_i | \hat{x}(t) | x_i, t_i \rangle = \frac{1}{N} \int_{x(t_i) = x_i}^{x(t_f) = x_f} \mathcal{D} x~x(t)e^{i S[x]}. $$

In the following the states and operators are meant to be understood in the Heisenberg picture. I know how to arrive at the rhs of the equation: Putting an identity into the lhs: $$ \langle x_f, t_i | \hat{x}(t) \int dx |x, t \rangle \langle x,t | | x_i, t_i \rangle \\ = \int dx \langle x_f, t_i |x, t \rangle x \langle x, t | x_i, t_i \rangle \\ = \int dx K(x_f, t_f, x, t) x K(x, t, x_i, t_i) \\ = \int dx \frac{1}{N_f} \int_{x(t) = x}^{x(t_f) = x_f} \mathcal{D} x e^{i S[x]} x \frac{1}{N_i} \int_{x(t_i) = x_i}^{x(t) = x} \mathcal{D} x ~ x(t)e^{i S[x]} \\ = \frac{1}{N} \int_{x(t_i) = x_i}^{x(t_f) = x_f} \mathcal{D} x x(t)e^{i S[x]}. $$

What is the meaning of this object? Can this be thought of an "expectation value" of some sort?

Of course, the argument is somewhat clear (as also mentioned in this question:For a given $x$, $x_i$ and $x_f$ you get (by $K(x_f, t_f, x, t)$ the probability-AMPLITUDE that a particle that was at $x_i$ at time $t_i$ will be at $x$ at time $t$, and with $K(x, t, x_i, t_i)$ the probability-amplitude that a particle starting at $x$ at time $t$ will be found at $x_f$ at time $t_f$.

So if you definitely know that the particle started at $x_f$ and ended at $t_f$ then the product of both yields the Amplitude by which the particle was at $x$ at time t. Integrating over all x might then give you what is widely called "expectation value" - except for some difficulties (that the cited question sadly doesn't resolve):

  • For the expectation value interpretation to hold, wouldn't we need the probability (absolute square of $K * K$? )

  • Can we be sure that the result is real? I tried to take the complex conjugate, but in order to proof that it will be the same result, I need to make additional assumptions on $K$

  • Usually expectation values are taken between the same states, but even between same states, we still have problems with expression not neccessarily being real.

Quantumwhisp
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    I would say an expectation value for some Hermitian operator $O$ would be $\langle \Psi | O | \Psi\rangle$ -- in other words, the state in the bra and ket are the same. If the bra and ket are different, like $\langle \Phi | O | \Psi \rangle$, I would call that a matrix element. Different people use different conventions , but off hand I don't recall hearing an expression like $\langle \Phi | O | \Psi \rangle$ referred to as an expectation value. It certainly is not real and can't be interpreted as an average value of $O$, so if you have seen this I think it is just sloppy language. – Andrew May 16 '22 at 01:07
  • Called an expectation value by whom? Which page? – Qmechanic May 17 '22 at 14:17
  • @Qmechanic it seems I missread the wiki page https://en.wikipedia.org/wiki/Path_integral_formulation#Expectation_values_and_matrix_elements

    More general, those objects seem to be called "matrix Elements", because of that I will try to reshape my question.

     – Quantumwhisp May 19 '22 at 16:01

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