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I am reading the first paper in Schwinger's QED anthology, where he discusses his action principle. In this, he writes down states that are simultaneous eigenkets of the field operators at all points of a space-like surface $\sigma $, $\lvert\Phi', \sigma\rangle$, which obey $\Phi(x)|\Phi', \sigma\rangle = \Phi'(x)|\Phi', \sigma\rangle$ for any $x \in \sigma$.

I am aware that $\Phi(x)$ will commute with $\Phi(y)$ if the separation between x and y is spacelike, and hence what he does is doable, at least in principle. But how does one write down such a state in terms of particle states? I have also seen this construction somewhere else (I think Peskin), where functional integrals are first introduced to deal with amplitudes of the form $\langle\Phi'_1, t_1\vert\Phi''_2,t_2\rangle$, i.e. the propagation amplitude between a simultaneous eigenket of all $\Phi(x,t_1)$ and a simultaneous eigenket of all $\Phi(x,t_2)$, and the formalism is then extended to deal with the vacuum in far past to vacuum in far future amplitude.

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  • $\begingroup$ @David Zaslavsky: Can you give me reference where this topic is clearly discussed? $\endgroup$ – rainman Apr 21 '13 at 13:24
  • $\begingroup$ @David Zaslavsky: Can you give me reference where this topic is clearly discussed? Is there only one eigenket $|\Phi, t \rangle $ associated with all the eigenvalues of $\hat{\Phi}$ ? Can you please state this clearly? It is making me crazy! $\endgroup$ – rainman Apr 21 '13 at 13:33
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    $\begingroup$ @far.westerner, You might consider 312006. As per my answer in the simple oscillator problem, you first construct position eigenstates in Fock space, and then generalize to QFT by simply tensoring an infinity of those. $\endgroup$ – Cosmas Zachos Feb 20 '17 at 17:59

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