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qft-El

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- TL;DR Summary
- There is a remark, which was added in the latest edition (2010) of Zee's book "Quantum field theory in a nutshell" which I do not understand. Such remark states that he's working in Heisenberg picture. I think anything there could be also interpreted in terms of Schrödinger picture, though.

Reading the introduction to path integrals given in the latest edition of Zee's "Quantum field theory in a nutshell", I have found a remark which I don't really understand. The author is evaluating the free particle propagator ##K(q_f, t; q_i, 0)##

$$\langle q_f\lvert e^{-iHt}\lvert q_i \rangle\underset{\delta t:=t/N}{=}\langle q_f\lvert e^{-iH\delta t}e^{-iH\delta t}...e^{-iH\delta t}\lvert q_i \rangle=\int\prod_{j=1}^{N=1} dq_i \langle q_{f}\lvert e^{-iH\delta t}\lvert q_{N-1}\rangle...\langle q_1\lvert e^{-iH\delta t}\lvert q_i\rangle$$

Using ##H=\frac{p^2}{2m}##, we can evaluate ##\langle q_{j+1}\lvert e^{-i\frac{p^2}{2m}\delta t}\lvert q_j\rangle\quad\forall j=1...N-1##.

Before doing that, I want to say that this amplitude can be interpreted both in the Schrödinger picture as evolving the state ##\lvert q_j\rangle## for a time ##t## and then computing the amplitude with the position eigenbra ##\langle q_{j+1}\lvert## or as the inner product between two instaneous eigenstates of ##\hat{q}(t)## in Heisenberg picture

$$\langle q_{j+1}, t_0+\delta t\lvert q_j, t_0\rangle=\langle q_{j+1}\lvert\hat{U}(t_0+\delta t)\hat{U}^{\dagger}(t_0)q_j, t_0\rangle=\langle q_{j+1}\lvert e^{-iH\delta t}\lvert q_j\rangle$$

After all, amplitudes should not depend on the picture used. Now, if we insert the (Schrödinger picture) completeness relation of momentum eigenstates

$$1=\int\frac{dp}{2\pi}\lvert p\rangle\langle p\lvert$$

we get

$$\langle q_{j+1}\lvert e^{-i\frac{p^2}{2m}\delta t}\lvert q_j\rangle=\int\frac{dp}{2\pi}\langle q_{j+1}\lvert e^{-i\frac{p^2}{2m}}\lvert p\rangle\langle p\lvert\lvert q_j\rangle=\int\frac{dp}{2\pi}e^{-i\frac{p^2}{2m}}\langle q_{j+1}\lvert p\rangle\langle p\lvert\lvert q_j\rangle$$

Then it's just a matter of evaluating this Gaussian integral. After this the author says: "

For the reasons above, I do not understand this remark:

$$\langle q_f\lvert e^{-iHt}\lvert q_i \rangle\underset{\delta t:=t/N}{=}\langle q_f\lvert e^{-iH\delta t}e^{-iH\delta t}...e^{-iH\delta t}\lvert q_i \rangle=\int\prod_{j=1}^{N=1} dq_i \langle q_{f}\lvert e^{-iH\delta t}\lvert q_{N-1}\rangle...\langle q_1\lvert e^{-iH\delta t}\lvert q_i\rangle$$

Using ##H=\frac{p^2}{2m}##, we can evaluate ##\langle q_{j+1}\lvert e^{-i\frac{p^2}{2m}\delta t}\lvert q_j\rangle\quad\forall j=1...N-1##.

Before doing that, I want to say that this amplitude can be interpreted both in the Schrödinger picture as evolving the state ##\lvert q_j\rangle## for a time ##t## and then computing the amplitude with the position eigenbra ##\langle q_{j+1}\lvert## or as the inner product between two instaneous eigenstates of ##\hat{q}(t)## in Heisenberg picture

$$\langle q_{j+1}, t_0+\delta t\lvert q_j, t_0\rangle=\langle q_{j+1}\lvert\hat{U}(t_0+\delta t)\hat{U}^{\dagger}(t_0)q_j, t_0\rangle=\langle q_{j+1}\lvert e^{-iH\delta t}\lvert q_j\rangle$$

After all, amplitudes should not depend on the picture used. Now, if we insert the (Schrödinger picture) completeness relation of momentum eigenstates

$$1=\int\frac{dp}{2\pi}\lvert p\rangle\langle p\lvert$$

we get

$$\langle q_{j+1}\lvert e^{-i\frac{p^2}{2m}\delta t}\lvert q_j\rangle=\int\frac{dp}{2\pi}\langle q_{j+1}\lvert e^{-i\frac{p^2}{2m}}\lvert p\rangle\langle p\lvert\lvert q_j\rangle=\int\frac{dp}{2\pi}e^{-i\frac{p^2}{2m}}\langle q_{j+1}\lvert p\rangle\langle p\lvert\lvert q_j\rangle$$

Then it's just a matter of evaluating this Gaussian integral. After this the author says: "

*we are***evidently**working in the**Heisenberg**picture." (page 11, 2010 edition)For the reasons above, I do not understand this remark:

- As I said one could work in both pictures, so why does he say that? Where is he
**evidently**using Heisenberg picture? - Furthermore, he seems to be using Schrödinger picture completeness relations, am I wrong?

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