The ambiguity function and the displacement operator basis in quantum mechanics

TL;DR

This paper introduces a novel method for calculating quantum operator expectation values using a displacement operator basis, providing an alternative to the Wigner-Weyl phase-space formalism.

Contribution

It presents a new c-function formalism based on the displacement operator basis, expanding the tools for quantum expectation value calculations beyond traditional phase-space methods.

Findings

Displacement operator forms a complete orthogonal basis for quantum operators.

The method connects to Wigner distribution and Weyl procedures.

Examples demonstrate the practical application of the formalism.

Abstract

We present a method for calculating expectation values of operators in terms of a corresponding c-function formalism which is not the Wigner--Weyl position-momentum phase-space, but another space. Here, the quantity representing the quantum system is the expectation value of the displacement operator, parametrized by the position and momentum displacements, and expectation values are evaluated as classical integrals over these parameters. The displacement operator is found to offer a complete orthogonal basis for operators, and some of its other properties are investigated. Connection to the Wigner distribution and Weyl procedure are discussed and examples are given.

Equations

$$\exp [\mathbf{A};\mathbf{B}]=\sum _{n=0}^{\infty }\frac{1}{n!}{\mathbf{A}}^n{\mathbf{B}}^n\ne \sum _{n=0}^{\infty }\frac{1}{n!}(\mathbf{AB}{)}^n.$$

$$A(\eta ,\xi )=\text{mbox}Tr\left\{\mathbf{A\Theta }(\eta ,\xi )\right\},$$

$$\mathbf{A}$$

$$\mathbf{\Theta }(\eta ,\xi )=\frac{e^{i(\eta \mathbf{q}+\xi \mathbf{p})/\hslash }}{\sqrt{2\pi \hslash }},$$

$$\text{mbox}Tr\left\{\mathbf{AB}\right\}=\iint \text{mbox}d\eta \text{mbox}d\xi A(\eta ,\xi )B(-\eta ,-\xi ).$$

$$\text{mbox}Tr\left\{\mathbf{\Theta }(\eta ,\xi )\mathbf{\Theta }(-{\eta }^{\prime },-{\xi }^{\prime })\right\}=\delta (\eta -{\eta }^{\prime })\delta (\xi -{\xi }^{\prime }),$$

$$\iint \text{mbox}d\eta \text{mbox}d\xi$$

$$=\delta (x-x^{\prime })\delta (k-k^{\prime }),$$

$$\frac{\partial }{\partial t}P(\eta ,\xi ,t)$$

$$\displaystyle\times H\left(\Big{.}\frac{\eta}{2}+\eta^{\prime},\frac{\xi}{2}+\xi^{\prime}\right)P\left(\Big{.}\frac{\eta}{2}-\eta^{\prime},\frac{\xi}{2}-\xi^{\prime},t\right)$$

$$=\frac{2}{\hslash }\iint \frac{\text{mbox}d{\eta }^{\prime }\text{mbox}d{\xi }^{\prime }}{\sqrt{2\pi \hslash }}\sin \frac{{\eta }^{\prime }\xi -\eta {\xi }^{\prime }}{2\hslash }$$

$$\times H({\eta }^{\prime },{\xi }^{\prime })P(\eta -{\eta }^{\prime },\xi -{\xi }^{\prime },t),$$

$$\frac{\partial }{\partial t}P(\eta ,\tau ,t)$$

$$\displaystyle i\hbar\frac{{\partial}{}}{{\partial}{t}}\mbox{Tr}\left\{{\boldsymbol{\rho}\mathbf{\Theta}}\right\}=\mbox{Tr}\left\{{\left(\Big{.}\mathbf{H}\boldsymbol{\rho}-\boldsymbol{\rho}\mathbf{H}\right)\mathbf{\Theta}}\right\}.$$

$$P(\eta ,\xi ,t)$$

$$\displaystyle=e^{iFt(2\xi-\eta t/m)/2\hbar}P\left(\Big{.}\xi-\frac{\eta}{m}t,\eta,0\right).$$

$$\frac{\text{mbox}d}{\text{mbox}dt}{e}^{\mathbf{B}(t)}={\int }_0^1\text{mbox}d\lambda e^{\lambda \mathbf{B}(t)}\frac{\text{mbox}d\mathbf{B}(t)}{\text{mbox}dt}{e}^{(1-\lambda )\mathbf{B}(t)},$$

$$\frac{\text{mbox}d}{\text{mbox}dt}P(\eta ,\xi )$$

$$\displaystyle\left[\Big{.}\eta\frac{{\mbox{d}}{\mathbf{q}(t)}}{{\mbox{d}}{t}}+\xi\frac{{\mbox{d}}{\mathbf{p}(t)}}{{\mbox{d}}{t}}\right]e^{i(1-\lambda)[\eta\mathbf{q}(t)+\xi\mathbf{p}(t)]/\hbar}\Big{\}}.$$

$$\mathbf{A}=\iint \text{mbox}d\eta \text{mbox}d\xi \mathbf{\Theta }(-\eta ,-\xi )\iint \frac{\text{mbox}dq\text{mbox}dp}{2\pi \hslash }\tilde{A}(q,p)\frac{e^{i(\eta q+\xi p)/\hslash }}{\sqrt{2\pi \hslash }}.$$

$$\tilde{A}(q,p)=\text{mbox}Tr\left\{\mathbf{AW}(q,p)\right\},$$

$$\mathbf{W}(q,p)=\frac{1}{\sqrt{2\pi \hslash }}\iint \text{mbox}d\eta \text{mbox}d\xi e^{-i(\eta q+\xi p)/\hslash }\mathbf{\Theta }(\eta ,\xi ).$$

$$A(\eta ,\xi )=\int \text{mbox}dq{e}^{-i\eta q/\hslash }⟨q-\xi /2\left|\mathbf{A}\right|q+\xi /2⟩,$$

$$\tilde{A}(q,p)=\int \text{mbox}d\xi e^{-ip\xi /\hslash }⟨q-\xi /2\left|\mathbf{A}\right|q+\xi /2⟩.$$

$$A(\eta ,\xi )=\delta (\xi )e^{i\eta x},$$

$$\tilde{A}(q,p)=\delta (q-x),$$

$$A$$

$$+\beta {\alpha }^{\ast }{e}^{i\eta x_1}\delta (x_2-x_1-\xi )+{\left|\beta \right|}^2{e}^{i\eta x_2}\delta (\xi )$$

$$={\left|\alpha \right|}^2{e}^{i\eta x_1}\delta (\xi )+{\left|\beta \right|}^2{e}^{i\eta x_2}\delta (\xi )+e^{i\eta (x_1+x_2-\xi )/2\hslash }$$

$$\displaystyle\times\left(\Big{.}\alpha\beta^{*}\delta(x_{1}-x_{2}-\xi)+\beta\alpha^{*}\delta(x_{2}-x_{1}-\xi)\right).$$

$$\displaystyle\psi(q)=\left(\Big{.}\frac{\Delta}{\pi}\right)^{1/4}\exp\left[\Big{.}-\frac{(q-x)^{2}}{2\Delta}-iqk/\hbar\right],$$

$$\mathbf{\rho }$$

$$\displaystyle P(\eta,\xi)=\frac{e^{i\eta x/\hbar+i\xi k/\hbar}}{\sqrt{2\pi\hbar}}\exp\left[\Big{.}-\frac{\eta^{2}}{4\hbar^{2}/\Delta}-\frac{\xi^{2}}{4\Delta}\right],$$

$$\displaystyle\widetilde{P}(q,p)=\frac{1}{\pi\hbar}\exp\left[\Big{.}-\frac{(q-x)^{2}}{\Delta}-\frac{(p-k)^{2}}{\hbar^{2}/\Delta}\right],$$

$$\mathbf{H}=\frac{{\mathbf{p}}^2}{2m}-F\mathbf{q},$$

Full text

The ambiguity function and the displacement operator basis in quantum mechanics

J. S. Ben-Benjamin1 and W. G. Unruh2

1Institute for Quantum Science and Engineering, Texas A&M University, Texas, USA.

2University of British Columbia, Vancouver, Canada.

We present a method for calculating expectation values of operators in terms of a corresponding c-function formalism which is not the Wigner–Weyl position-momentum phase-space, but another space. Here, the quantity representing the quantum system is the expectation value of the displacement operator, parametrized by the position and momentum displacements, and expectation values are evaluated as classical integrals over these parameters. The displacement operator is found to offer a complete orthogonal basis for operators, and some of its other properties are investigated. Connection to the Wigner distribution and Weyl procedure are discussed and examples are given.

Wolfgang Schleich is a master of quantum mechanics in phase-space, and of physics in general. He is always full of insight and excels in finding simplicity. We believe we have found an interesting way to do quantum mechanics in phase-space, a field in which Prof. Schleich has made enormous contributions, and tried investigating its source to the bottom, in the spirit of Prof. Schleich. We are elated to dedicate these results to him.

1 Introduction

In the early 1930’s, Wigner pioneered the phase-space formulation of quantum mechanics, introducing the Wigner distribution [1]. He did so in an attempt to make quantum mechanics more classical (statistical mechanics) looking, and indeed, to calculate some quantum properties of gasses. Since then, other phase-space formulations of quantum mechanics were created. Besides to construct a quantum theory of statistical mechanics, the drive to create phase-space distributions was due to several aspects: One was a fundamental aspect – trying to create new formulations of qunatum mechanics and studying the uncertainty principle; another, not too distant motivation was the study of the classical–quantum interface; still other reasons are for mathematical, as well as conceptual simplicity. Indeed, Wolfgang Schleich is notorious for utilizing the Wigner distribution to simplify physical problems and to give insight into their inner-workings [2].

Independently of these developments, there was a push to determine which operators should be used in quantum mechanics to describe systems. That is, starting with a system which we might describe classically by some quantity $A_{\mbox{\scriptsize{C}}}(q,p)$, what is the quantum analog? Due to the quantum commutation relations between the position and momentum operators, it is far from obvious how to obtain the ‘quantum version’ of $A_{\mbox{\scriptsize{C}}}(q,p)$. Several answers were given, including one by Weyl [3]. In the late 1940’s, Moyal realized the connection [4] between Weyl’s procedure and Wigner’s distribution. Namely, that111We denote operators by boldface characters throughout.$\mbox{Tr}\left\{{\boldsymbol{\rho}\mathbf{A}}\right\}=\iint\mbox{d}q\mbox{d}p\,\tilde{P}(q,p)\tilde{A}(q,p)$, if $\tilde{P}$ is Wigner’s distribution coming from $\boldsymbol{\rho}$ and $\mathbf{A}$ is the operator coming from $\tilde{A}$ vie Weyl’s procedure. In the mid 1960’s, Cohen found the connection between all of the distributions and the operators, and gave a way to generate arbitrary distributions [5].

We present the theory behind one such distribution, the ambiguity distribution, which is seldom used in quantum mechanics, and present many of its properties and the properties of its accompanying classical operator, including the $\mathbf{A}\leftrightarrow A$. We also show that it has multiple advantages over the widely-used distributions.

Royer [6] has shown that the Wigner distribution [1, 9, 2] is the expectation value of a seed operator222 The semicolon (;) in the exponent in Eq. (26) means that the entire exponential operator is ordered in the sense that [7, 8]

$\displaystyle\exp[\mathbf{A};\mathbf{B}]=\sum_{n=0}^{\infty}\frac{1}{n!}\mathbf{A}^{n}\mathbf{B}^{n}\neq\sum_{n=0}^{\infty}\frac{1}{n!}(\mathbf{A}\mathbf{B})^{n}.$

(1)

$\mathbf{W}(q,p)=2\exp[2i(\mathbf{p}-q);(\mathbf{q}-q)/\hbar]/\hbar$, also called the “displaced parity operator,” and Englert [7] has found the above elegant expression for $\mathbf{W}$ in terms of operator-ordered exponentials. We [10] have reviewed these expressions for $\mathbf{W}$ and found some additional ones, and showed how operator-ordering emerges naturally from the seed operator. It is in general interesting to see how such distributions come about [10, 11].

Now we find that the displacement operator, which we denote by $\mathbf{\Theta}$, is another such seed operator; not for Wigner functions, rather for their characteristic function, also known as the ambiguity function. We thus call $\mathbf{\Theta}$ the ambiguity seed-operator. This operator has some attractive properties, such as operator orthogonality [Eq. (6)] and operator completeness [Eq. (7)], and could be used for phase-space quantum mechanics [Eq. (5)]. As we show in Sec. 4, it happens that $\mathbf{\Theta}$ is intimately-related to the Wigner seed operator $\mathbf{W}$. This formalism could be used for mathematical manipulations of operators, and it was used in [12] for studying the decoherence of a harmonic oscillator in a heat bath, and in [13, 14], for studying fundamental issues in relativistic quantum field theory. Already in [15] it was suggested that the ambiguity function could be used for quantum mechanics, but as far as we know, the formalism which we present in the next section is new.

Compared to the Wigner distribution, our new formalism has some advantages and some disadvantages. In our formalism, calculation of the expectation value of (polynomial) operators does not involve any integration – just derivatives and multiplication (as in Eq. (38). This is a clear advantage over the Wigner distribution. Also, in some cases, e.g. in [12], the distribution we present is easier to evolve in time. The Wigner c-number corresponding to a Hermitian operator is real, while in our formalism it need not be. This may or may not be advantageous – for example in some cases, obtaining $A(\eta,\xi)$ from $\mathbf{A}$ might be easier than obtaining $\widetilde{A}$ from $\mathbf{A}$. Like the Wigner distribution, the ambiguity function transforms nicely under canonical transformations. In contrast with the Wigner distribution, the ambiguity distribution does not satisfy the marginals.

2 c-numbers for use in quantum mechanics

We present a method for obtaining the c-number distribution (ambiguity function) $A(\eta,\xi)$ for any arbitrary operator $\mathbf{A}$, and show how it can be used for calculating quantum mechanical expectation values. Like the Wigner function [1, 9, 2, 16], this c-number distribution is also obtained from the expectation value of some seed operator, $\mathbf{\Theta}(\eta,\xi)$, the ambiguity seed operator. Starting with an arbitrary operator $\mathbf{A}$, we define the c-number

[TABLE]

from which the operator $\mathbf{A}$ could be recovered by333All integrations range from $-\infty$ to $\infty$ unless otherwise specified.

[TABLE]

where the operator $\mathbf{\Theta}(\eta,\xi)$ is

[TABLE]

which is known as the displacement operator (divided by $\sqrt{2\pi\hbar}$). As we show in Sec. 4, the c-number $A(\eta,\xi)$ in Eq. (2) is the double Fourier transform, or characteristic function, of the Wigner function $\widetilde{A}(q,p)$ corresponding to the operator $\mathbf{A}$, which is also known as the ambiguity function in the field of radar [17, 18]. We note that $\eta$ and $\xi$ have dimensions of momemtum and position, respectively.

We may use our definition in Eq. (2) for the c-number functions in order to compute quantum mechanical expectation values, or more generally, traces of operator products. We find that the trace of the product of two arbitrary operators, $\mathbf{A}$ and $\mathbf{B}$, could be computed by

[TABLE]

The expectation value of $\mathbf{A}$ is obtained when $\mathbf{B}$ is the density matrix $\boldsymbol{\rho}$.

As we show in Apps. A and B, Eqs. (2), (3), and (5) are consequences of the facts that the trace

[TABLE]

and that

[TABLE]

which we derive in App. A.

3 Time evolution

We may use this formulation to evolve the quantum state $P(\eta,\xi)=\mbox{Tr}\{\boldsymbol{\rho}\mathbf{\Theta}\}$ in time using the Schrödinger (von Neumann) equation, or to evolve an arbitrary operator $A(\eta,\xi)=\mbox{Tr}\{\mathbf{A}\mathbf{\Theta}\}$ in time using the Heisenberg equation. In particular, we give a prescription purely in terms of ambiguity quantities.

[TABLE]

for the quantum state. To evolve an arbitrary operator, $A(\eta,\xi)$, in time, take $t\longrightarrow-t$ and replace $P$ by $A$ in Eq. (9). For example, (using Eqs. (9) and (35)) time evolution under the constant force Hamiltonian is

[TABLE]

which could also be obtained from

[TABLE]

The solution to Eq. (10) is

[TABLE]

That is, the ambiguity function of the quantum state is evolved to some time $t$ by simple substitution at time $t=0$ and multiplication be a phase.

In Ref. [12], the time evolution enters through the $\mathbf{\Theta}$ operator. Particularly, $\mathbf{\Theta}(\eta,\xi,t)$ is found by using the time-evolution of the position and momentum operators. Using the formula for the derivative of the exponential of an operator

[TABLE]

it was found that

[TABLE]

4 Connection to Wigner functions and to the Weyl procedure

The Weyl procedure is a procedure introduced by Weyl for obtaining quantum operators $\mathbf{A}$ from c-numbers $A_{\mbox{\scriptsize{W}}}$ (which Moyal showed that $A_{\mbox{\scriptsize{W}}}$ is the Wigner function $\tilde{A}$ corresponding to $\mathbf{A}$ [4]). Weyl was interested in determining what quantum operators one should use given classical analog444 There is also interest in the inverse procedure – obtaining the classical function $\widetilde{A}(q,p)$ from the operator $\mathbf{A}$ [20, 21]. . He proposed [3, 16]

[TABLE]

Therefore, comparing Eqs. (3) and (16), we find that the c-number $A$ which we defined in Eq. (2) is the double Fourier transform of the Wigner function $\widetilde{A}$ of $\mathbf{A}$.

An interesting connection also exists between the seed operator for the ambiguity function, $\mathbf{\Theta}(\eta,\xi)$ and the seed operator for the Wigner function $\mathbf{W}(q,p)$. The Wigner function $\widetilde{A}$ is obtained from $\mathbf{A}$ via [7, 10]

[TABLE]

and $\mathbf{\Theta}(\eta,\xi)$ is connected to $\mathbf{W}(q,p)$ by Fourier transform (see App. C)

[TABLE]

So we see it is no accident that also $A(\eta,\xi)$ and $\widetilde{A}(q,p)$ are related by Fourier transform.

Another interesting connection could be found when calculating $A$ and $\widetilde{A}$ from $\mathbf{A}$. In the position representation, the ambiguity function is

[TABLE]

while the Wigner–Weyl function is

[TABLE]

5 Examples

Ex: Position states. Because $A(\eta,\xi)$ is complex, it describes the position state $\mathbf{A}=\left|{x}\middle\rangle\middle\langle{x}\right|$ as a function with depdendence on both $\eta$ and $\xi$,

[TABLE]

in contrast to the Wigner function of $\mathbf{A}$,

[TABLE]

which has no momentum dependence. The superposition of two positions $\left|{\psi}\right\rangle=\alpha\left|{x_{1}}\right\rangle+\beta\left|{x_{2}}\right\rangle$ is

[TABLE]

Ex: Gaussian state. The density matrix of the Gaussian state wavefunction peaked about the position $q=x$, and having average momentum $k$,

[TABLE]

is

[TABLE]

corresponding to the ambiguity function

[TABLE]

and to the Wigner function

[TABLE]

where the semicolon (;) in the exponent is Schwinger ordered-exponential notation, as in Eq. (1).

Ex: Constant force Hamiltonian. The constant force Hamiltonian

[TABLE]

corresponds to the Wigner function

[TABLE]

To find the ambiguity function associated with this Hamiltonian, we use that

[TABLE]

and that

[TABLE]

which are like Bopp operators [22], which could be used for obtaining Wigner functions from operators in a simple way [10]. Using Eqs. (31), the ambiguity function $H(\eta,\xi)$ is

[TABLE]

and the expectation value of the Hamiltonian is calculated via Eq. (5)

[TABLE]

where in the last equality in Eq. (38), the expression is evaluated at $\eta=\xi=0$, and $P(\eta,\xi)$ is the ambiguity function which is obtained from the state’s density matrix via Eq. (2). Interestingly, it is the value of the operated-on quantum state $P(\eta,\xi)$ at zero displacement ($\eta=\xi=0$) that gives the expectation value of the operator. Amazingly, no integrals are required; this expression involves only derivatives and multiplication. This is generally the case for operators which are polynomial in $\mathbf{q}$ and $\mathbf{p}$.

6 Properties of $\mathbf{\Theta}(\eta,\xi)$

We now discuss some properties of the ambiguity seed operator.

The ambiguity seed operator has the property that its form is unchanged under a canonical transformation

[TABLE]

that is to say, the new parameters become

[TABLE]

Specifically,

[TABLE]

where $\mathbf{\Theta}_{\mathbf{Q},\mathbf{P}}(\eta,\xi)$ is the operator $\exp[i(\eta\mathbf{Q}+\xi\mathbf{P})/\hbar]/\sqrt{2\pi\hbar}$. If the Jacobian $J$ of the transformation in Eq. (40) is unity, that is, $J=\alpha\delta-\beta\gamma=1$, then the commutators $\left[{\mathbf{q}}\,,{\mathbf{p}}\right]=J[\mathbf{Q},\mathbf{P}]$ are equal (which makes the transformation canonical), and we have the symmetry that

[TABLE]

where we have used Eqs. (3) and (41), and $\eta$ and $\xi$ instead of $-\eta$ and $-\xi$. Using Eq. (6), we find that

[TABLE]

which means that if we calculate $A_{\mathbf{q},\mathbf{p}}(\eta,\xi)$ corresponding to some operator $\mathbf{A}(\mathbf{q},\mathbf{p})$, then we can obtain the c-number $A_{\mathbf{Q},\mathbf{P}}(\eta,\xi)$ corresponding to $\mathbf{A}(\mathbf{Q},\mathbf{P})$ via the simple substitution, Eq. (40)

[TABLE]

The trace of $\mathbf{\Theta}(\eta,\xi)$ is

[TABLE]

and its integral is

[TABLE]

which is the parity operator [7], where the semicolon (;) in the exponent has the same meaning as it does in Eq. (26) [Eq. (1)]. This is not shocking, because it offers another connection to the Wigner distribution which is the expectation value of the displaced parity operator [6].

Integrating the ambiguity seed over one variable, we get

[TABLE]

which means that

[TABLE]

Composition rule. Since the ambiguity seed operators are phase-space displacement operators, we would expect that they could be combined into a different displacement. This is true, however, up to a phase

[TABLE]

which comes from the Campbell-Baker-Hausdorff relation. However, when actually displacing an operator555 The ‘$f$’ is boldface, indicating that operator-ordering is important. $\mathbf{f}(\mathbf{q},\mathbf{p})$, the phase cancels

[TABLE]

7 Conclusions

We have presented a formalism for obtaining c-function distributions corresponding to quantum states and to quantum operators, and showed how to use them for calculation of expectation values. Surprisingly, in contrast with the usual phase-space formalisms (Wigner, Kirkwood–Rihaczek, etc.), expectation values of operators do not involve any integration – only derivatives. These distributions were shown to posess attractive features, such as symmetries under canonical quantum transformations. There are prospects for generalizing this treatment using ideas from Refs. [5, 23].

8 Acknowledgements

This paper is warmly dedicated to Wolfgang Schleich.

The authors would like to thank the organizers of this special issue for allowing them to participate, and would like to thank Profs. M. Scully and L. Cohen for their valuble comments and suggestions. In addition, JSB would like to thank the Robert A. Welch Foundation (Grant No. A-1261), the Office of Naval Research (Award No. N00014-16-1-3054), and the Air Force Office of Scientific Research (FA9550-18-1-0141) for their the support, and WGU would like to thank NSERC Canada (Natural Science and Engineering Research Council), the Hagler Fellowship from HIAS (Hagler Institute for Advanced Studies), Texas A&M University, CIfAR, and the Humbolt Foundation for support.

Appendix A Proof of the relationship between Eqs. (2) and (3)

To see that Eqs. (2) and (3) are correct, we insert one into the other to check for consistency. First, To verify Eq. (3), we consider the position–momentum matrix elements of the arbitrary operator $\mathbf{A}$

[TABLE]

Using Eq. (7), we find that Eq. (54) is self-consistent

[TABLE]

The final check of the consistency of Eqs. (2) and (3) involves inserting Eq. (3) into Eq. (2)

[TABLE]

and using Eq. (6), we find that Eq. (57) is self-consistent

[TABLE]

Using the position–momentum and momentum–position matrix elements of the ambiguity seed operator

[TABLE]

which come from the Campbell-Baker-Hausdorff relation [19]

[TABLE]

we can obtain Eqs. (6) and (7) in a straight-forward manner; the results, Eqs. (2) and (3) of course, are basis independent.

Appendix B Proof of our expectation value method

Eq. (5) is obtained from the same principles. Taking Eq. (2) for $\mathbf{A}$ and $\mathbf{B}$, we take the trace and use Eq. (6) to find that

[TABLE]

which is Eq. (5), where we have used Eq. (6).

Appendix C Relationship between Ambiguity and Wigner seeds

In Eq. (18), we stated that the ambiguity function seed operator, Eq. (4) and the Wigner seed operator,

[TABLE]

(where the semicolon denotes Schwinger exponent ordering, as we mention in a footnote on the first page – see Eq. (1).) are related via Fourier transform. Here we provide the proof. Starting With Eq. (18) (reproduced here for clarity),

[TABLE]

where we used Eq. (60). Integrating over $\xi$, we have

[TABLE]