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In Dysons Book "Quantum Field Theory", In the Section on Schwinger's action principle, he introduces Feynman's path integral method to obtain transition amplitudes:

$$ \langle \phi_2, t_2 | \phi_1, t_1 \rangle = \Sigma_H N e^{\frac{i}{\hbar}I_H} $$ H is a classic path of $\phi(t)$ beginning at $t_1$ with value $\phi_1$ and ending at $t_2$ with value $\phi_2$. The sum goes over all classical paths that satisfy these requirements, and $N$ is a normalization factor.

Here, $\langle \phi_2, t_2 |$ is supposed to be an eigenstate of the field operator $\hat{\phi}(t_2)$ with eigenvalue $\phi_2$ at a space like plane in the 4 dimensional spacetime for simplicity I choose the plane with $x^0 = t_2$, and thus wrote $t_2$ for simplicity.

In the same manner $| \phi_1, t_1 \rangle$ is an eigenstate at time $t_1$ (By "eigenstate at $t_1$", I want to express that it's an eigenstate of the operator at $t_1$, so the operator is $\hat{\phi}(t_1)$). Note that because the operator can change in time, $\langle \phi_1, t_1 | \phi_1, t_2 \rangle$ isn't necessarily 1.

I assume he is working in the Heisenberg picture here, because if he weren't, there should be a time evolution operator in between the two states, which apparently isn't there.

Afterwards, Dyson introduces matrix elements as an additional assumption: $$ \langle \phi_2, t_2 | \hat{o}(t)|\phi_1, t_1 \rangle = \Sigma_H N O[\phi_H(t)] e^{\frac{i}{\hbar}I_H} $$ Where $O[\phi_H(t)]$ is the classical observable, evaluated at time $t$ at field-value $\phi_H(t)$.

My Question: Can I derive this formula from the first formula given, that deals solely with the transition amplitudes of the states?

I tried to do so, but I always fail at some point: I assume completenes of the eigenvectors ${ |\phi_i \rangle }$ expand the $\mathbb{1}$ operator and put it in

$$ \begin{array}{ccl} \langle \phi_2, t_2 | \hat{o}(t)|\phi_1, t_1 \rangle & = & \langle \phi_2, t_2 | \mathbb{1} \hat{o}(t) \mathbb{1} |\phi_1, t_1 \rangle \\ & = & \Sigma_{i,j} \langle \phi_2, t_2 |\phi_i, t \rangle \langle \phi_i, t | \hat{o}(t) |\phi_j, t \rangle \langle \phi_j, t |\phi_1, t_1 \rangle \\ & = & \Sigma_{i,j} \langle \phi_i, t | \hat{o}(t) |\phi_j, t \rangle \ \Sigma_{H_1} N_1 e^{\frac{i}{\hbar}I_{H_1}} \Sigma_{H_2} N_2 e^{\frac{i}{\hbar}I_{H_2}} \end{array} $$ Here $H_1$ is a path that begins at time $t_1$ with value $\phi_1$, and ends at time $t$ with value $\phi_j$, same goes for $H_2$. If ${ |\phi_i \rangle }$ would be a set of eigenvectors of Operator $\hat{O}$, I'd be done:

$$ \Sigma_{i} \langle \phi_i, t | \hat{o}(t) |\phi_i, t \rangle \ \Sigma_{H_1} N_1 e^{\frac{i}{\hbar}(I_{H_1}+I_{H_2})} \Sigma_{H_2} N_2 =\Sigma_H N O[\phi_H(t)] e^{\frac{i}{\hbar}I_H} $$ Where I used that $I_{H_1} + I_{H_2} = I_H$, if $H_1$ ends at $\phi_i$ and $H_2$ begins at $\phi_i$, and I assumed $\langle \phi_i, t | \hat{o}(t) |\phi_i, t \rangle = O[\phi_H(t)]$.

I however fail to derive this equation to hold if the states are no eigenvectors of $\hat{O}$, and thus the question.

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The fields $\phi^{\alpha}$ appearing in Dyson's book describe canonical coordinates whose corresponding quantum operators commute by definition at equal times: $$[\hat{\phi}^{\alpha}(r_1, t), \hat{\phi}^{\beta}(r_2, t)] = 0 $$ Operators $\hat{O}$ which are not diagonal in the coordinate basis should necessarily be functions of both the canonical coordinates and canonical momenta: $\hat{\pi}_{\alpha}(r, t)$

In order to compute matrix elements of this kind of operators, it is almost inevitable to use phase space path integrals, which are more general than the coordinate of configuration space path integrals.

This type of path integrals are not very popular in quantum field theory, where the interesting operators $\hat{O}$ are usually polynomials in the field variables, thus diagonal in the field theory configuration space. In addition, both the configuration and phase spaces are infinite dimensional in this case, which makes the problem even harder.

However, in quantum mechanics, there is a quite developed phase space path integration theory. Let me use here the symbols $q^{\alpha}$ and $p^{\beta}$ for the canonical coordinates and momenta as customary in quantum mechanics.

So the problem that I'll address is the computation of a matrix element of an operator depending on both the canonical coordinates and the canonical momenta between diffrent time coordinate eigenstates. This operator is of course non-diagonal in the coordinate basis.

Since, there are many quantum operators which correspond to the same classical function in phase space, the integration cannot in general performed on the operator by a simple replacement of the canonical operators by the corresponding functions in phase space. Secondly, the result will depend on the discretization method of the path integral in the computation of the matrix element.

It turns out that the solutions to both problems described above are connected, i.e., there are operator ordering schemes and corresponding discretization schemes for which the computation of the matrix element by means of the path integral will be correct. Please see the following thesis by Valtakoski, where this problem is analyzed in detail. For example, if we use the Weyl ordering and discretize the path integral at the midpoints we get correct results (table 3.2 in the thesis).

Response to follow up question

On the contrary, the ordering problem exists, and is much harder in quantum field theory. I only stated that generally people are more interested in matrix elements of field configurations. I preferred considering quantum mechanical path integrals because rigorous results exist and the path integral result can be compared to other methods such as canonical quantization.

In quantum field theory, when the operators depend on the field configurations and their momenta, then operator ordering and discretization schemes influence the results; and in general the problem can be much harder due to the infinite dimensionality of the phase space and also the lack of other quantization methods to compare with.

In the given example: the derivative $\partial_{\mu}\phi$, the time derivative: $\partial_{0}\phi$ does not commute at equal times with the field configurations (unlike the spatial derivatives $\partial_{1,2,3}\phi$ which do commute). For operators containing this derivative, one can still use the Weyl ordering scheme and the midpoint discretization because the scalar field phase space although infinite dimensional can be quantized according to the rules of canonical quantization.

When the theory contains fermions, then already the complex conjugate spinors do not commute at equal times with the fermionic configurations. Here there is a need of fermionic operator ordering and fermionic discretization rules to evaluate the path integral correctly, please see Bastianelli and van Nieuwenhuizen's review .

For gauge fields the problem is even more difficult, because their phase space is not flat due to the constraints and I don't think that any rigorous results about the ordering and the discretization even exist.

Finally, please notice that naively replacing a non-diagonal operator by a classical symbol in the path integral will almost surely produce results that are correct only semiclassically, because the ordering introduces errors of the order of $\hbar$ due to the non-vanishing commutators.

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  • $\begingroup$ I don't quite understand why in QFT there's no Problem, since in QFT since the interesting operators most times don't just contain the field variables, but also their "derivations" $\widehat{\partial_\mu \phi}$. Are those also diagonal in the "field-variable-basis"? $\endgroup$ – Quantumwhisp Jul 2 '17 at 18:37
  • $\begingroup$ @Quantumwhisp I have added an answer in the main text to your additional question $\endgroup$ – David Bar Moshe Jul 3 '17 at 8:46

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