Statistics on Lefschetz thimbles: Bell/Leggett-Garg inequalities and the classical-statistical approximation

While, in quantum theory, all states are equal, some states are more equal than others. If we do the statistics correctly, we do not really need to include all states in order to get an accurate enough result. This is the idea behind the stochastic method. The question is: What determines whether one state is more important than another, or, in other words, from which Probability Distribution Function (PDF) should we sample these states? The answer is well known in QFT and it can be extracted from a path-integral approach to the problem.

Recall that, to calculate the expectation value $\braket{\hat{\mathcal{O}}(t)}$ of some operator $\hat{\mathcal{O}}(t)$, we compute $\braket{\hat{\mathcal{O}}}={\rm Tr}\big{[}\hat{\rho}(t_{0})\hat{\mathcal{O}}(t)\big{]}/{\rm Tr}\big{[}\hat{\rho}(t_{0})\big{]}$, where we assume the density operator $\hat{\rho}$ is known at the time $t_{0}$. The partition function ${\rm Tr}\left[\hat{\rho}(t_{0})\right]$ is the integral of the PDF. For an analogy, consider the normalization $\int\mathrm{d}x\;e^{-x^{2}}$, arising when calculating $\braket{x^{2}}=\int\mathrm{d}x\;e^{-x^{2}}x^{2}/\int\mathrm{d}x\;e^{-x^{2}}$ for the Gaussian PDF $e^{-x^{2}}$. In the expression for $\langle\hat{\mathcal{O}}(t)\rangle$, the initial density operator $\hat{\rho}(t_{0})$ and the operator $\hat{\cal O}(t)$ are evaluated at different times. Therefore, to construct a path-integral representation of this expectation value, we should connect these operators together by constructing a closed-time path integral, according to the Schwinger-Keldysh prescription Schwinger:1960qe ; Keldysh:1964ud . More specifically, we can write the expectation value as

$\displaystyle\langle\hat{\cal O}(t)\rangle=\frac{\int{\mathcal{D}}\phi\langle\phi^{+}_{0};t_{0}|\hat{\rho}(t_{0})|\phi^{-}_{0};t_{0}\rangle e^{\frac{i}{\hbar}\int_{\cal C}\mathrm{d}t^{\prime}\,L}{\mathcal{O}}(t)}{\int{\mathcal{D}}\phi\langle\phi^{+}_{0};t_{0}|\hat{\rho}(t_{0})|\phi^{-}_{0};t_{0}\rangle e^{\frac{i}{\hbar}\int_{\cal C}\mathrm{d}t^{\prime}\,L}}~{},$

(2)

where $\mathcal{D}\phi\equiv\mathcal{D}\phi^{+}\,\mathcal{D}\phi^{-}$ is the functional integration measure, $L$ is the Lagrangian and $\mathcal{O}(t)$ is the field-configuration representation of the operator. The ${\mathcal{C}}$ on the time integrals indicates the closed-time contour: a contour with a time-ordered “$+$” branch, starting from $t_{0}$ and ending at some time $t_{m}>t>t_{0}$, and an anti-time-ordered “$-$” branch running backwards from $t_{m}$ to $t_{0}$. The superscripts “$+$” and “$-$” on the initial field configuration $\phi_{0}$ identify whether its time argument lies on the $+$ or $-$ branch of the closed-time contour $\mathcal{C}$.

Following the Schwinger-Keldysh approach, the expression (2) can be obtained directly by using the time-evolution operator $\hat{U}(t_{1},t_{2})=\mathrm{T}\exp\big{[}-\frac{i}{\hbar}\int_{t_{1}}^{t_{2}}\mathrm{d}t^{\prime}\,\hat{H}(t^{\prime})\big{]}$, where $\hat{H}$ is the Hamiltonian operator and $\mathrm{T}$ is the time-ordering operator,¹¹1In the absence of external sources, the Hamiltonian operator is time-independent in the Heisenberg picture. Here, we have allowed for explicit time dependence, e.g., through the parameters of the theory. or by inserting complete sets of eigenstates of the Heisenberg-picture field operators. We note that the choice of the end time of the contour $t_{m}$ is arbitrary, so long as $t_{m}>t$. Alternatively, one can arrive at the same expression from the functional Schrödinger equation (1) by recalling that its solution can be written as a convolution of the path integral with the initial wavefunctional, i.e., $\Psi(\phi_{m};t_{m})=\int^{\phi_{m}}{\mathcal{D}}\phi\,e^{iS/\hbar}\Psi(\phi_{0};t_{0})$, with the expectation value computed as $\braket{\hat{\mathcal{O}}(t)}=\int{\mathcal{D}}\phi\,\Psi^{*}(\phi_{t}^{-};t)\mathcal{O}(t)\Psi(\phi_{t}^{+};t)$.

With this path integral in mind, our plan to compute expectation values by picking a sample of important states is analogous to doing an integral using generated samples, such as one might do in a Monte-Carlo approach. However, by rephrasing QFT in the language of statistics, we are not led to any direct solution. In fact, we face a more challenging question than is found in standard statistics: How to deal with the fact that the PDF is complex-valued.

The non-negativity of the PDF in standard statistical analysis has a number of crucial effects. On one hand, non-negativity is an important condition to guarantee a well-behaved Markov chain, thereby allowing a Monte Carlo computation. On the other hand, there are some key properties that are implied by non-negativity. One of them relates to the Bell inequalities Bell:1964kc . For instance, from the viewpoint of statistics, there can be no violation of the inequality

$\displaystyle-3\leq\langle AB\rangle+\langle BD\rangle-\langle AD\rangle\leq 1~{},$

(3)

if the observables $A$, $B$ and $D$, of value $\pm 1$, are drawn from some real-valued, non-negative distribution function. One can prove this by generating samples according to such a distribution. Since any single realisation will satisfy the inequality, so will the average, given that $\braket{AB}=\sum_{i=1}^{N}A_{i}B_{i}/N$ and $A_{i}$, $B_{i}$ and $D_{i}$ are restricted to be $\pm 1$. However, a complex PDF can lead to violations of such inequalities.

Complex PDFs call for complex analysis. Consider, e.g., the complex Gaussian integral $\int\mathrm{d}x\,e^{ix^{2}}$. With the complex PDF $e^{ix^{2}}$, we cannot ascertain which states are more probable than others, unlike with the normal distribution $e^{-x^{2}}$. Nevertheless, we know that we can perform the integral by deforming the contour of integration into the complex plane by Cauchy’s theorem. By choosing a better contour, we can then make sense of the statistics. In this simple case, we find a normal distribution if we rotate the integration contour by $\pi/4$ in the complex $x$ plane, so that $ix^{2}\to-x^{2}$. Under the same principle, the Lefschetz thimble approach provides a powerful tool for QFT, whereby we complexify all real-valued fields Witten:2010cx and are furnished with a prescription for finding a suitable integration contour. The application of the Lefschetz thimble approach to real-time path integrals has recently attracted significant attention Tanizaki:2014xba ; Cherman:2014sba ; Alexandru:2016gsd ; Alexandru:2017lqr ; Mou:2019tck ; Ai:2019fri ; Mou:2019gyl ; Alexandru:2020wrj .

A popular approximation used in the evaluation of the real-time path integral is the Classical-Statistical (CS) approximation. There, the quantum evolution is replaced by the classical evolution of an ensemble of initial conditions, drawn from a non-negative PDF. The interpretation in the context of the Lefschetz thimble Monte Carlo is that we keep only the critical points of the action and allow only for non-negative Wigner functions. This is quantitatively a good approximation in the limit where field occupation numbers are large. Its validity may be further expressed in terms of discarding diagrams in a perturbative expansion (at high temperature Aarts:1997kp ), in terms of the “statistical” propagator being much larger than the “spectral” propagator (in direct comparison with quantum evolution based on truncations of the 2PI effective action Aarts:2001yn ; Arrizabalaga:2004iw ; Rajantie:2006gy ; Berges:2007ym ; Berges:2010nk ) or in terms of the state being “squeezed” (e.g., in cosmological perturbations during inflation Mukhanov:1990me ).

The prescription is as follows: When field modes acquire large occupation numbers, one may evolve them using classical equations of motion and compute observables as in classical field theory. When the fields interact, all modes are coupled, but the dynamics is still expected to be dominated by the ones with high occupancy. In a few specific cases, one may opt to seed the evolution with an initial state resembling the quantum zero-point fluctuations (the “half” Rajantie:2000nj ; GarciaBellido:2002aj ; Smit:2002yg , and for critical analyses, see Refs. Braden:2018tky ; Hertzberg:2020tqa ). This assumes that the evolution is linear (non-interacting) until the phenomenon under consideration (e.g., resonance, instability or inflation) amplifies these seeds into large occupation numbers, $\gg 1$. The “half” can of course not lead to truly quantum phenomena and is, from the point of view of the classical evolution, simply a curious non-thermal initial condition. Even so, it can give rise to spurious effects, since it has divergent energy in the UV (see, for instance, Ref. Arrizabalaga:2004iw ).

In the case of a free theory, the CS approximation is, in some sense, exact. In spite of this, the aim of this article is to describe where and why the CS approximation is nevertheless incapable of describing the violation of Bell-type inequalities, both spatial Bell inequalities and temporal Leggett-Garg inequalities, even for a free theory.

The remainder of this article is organised as follows. In Sec. 2, we provide a brief introduction to the CS approximation within the context of the Lefschetz thimble approach. In Secs. 3 and 4, we focus on the temporal Leggett-Garg inequalities and the spatial Bell inequalities, respectively. Our conclusions are presented in Sec. 5. Some useful results are collected in the Appendices.

To compute the expectation value appearing in Eq. (2), it proves convenient to introduce the following field variables:²²2We adopt the coefficients used in Refs. Greiner:1996dx ; Aarts:1997kp but the notation used in Ref. Kamenev:2009 . For clarity, we suppress the spacetime index for the fields, except in the places where the specific spacetime sites are required.

$\displaystyle\phi^{\mathrm{cl}}\equiv\frac{1}{2}\left(\phi^{+}+\phi^{-}\right)~{},\qquad\phi^{\mathrm{q}}\equiv\phi^{+}-\phi^{-}~{}.$

(4)

In terms of these variables, the closed-time path integral involves the exponent

$\displaystyle\frac{i}{\hbar}\int_{\cal C}\mathrm{d}t\;L$

$\displaystyle=\frac{i}{\hbar}\int_{\partial{\mathcal{C}}}\mathrm{d}^{d}\mathbf{x}\left[-\phi_{0}^{\mathrm{q}}\dot{\phi}_{0}^{\mathrm{cl}}\right]-\mathcal{I}~{},$

(5)

with the bulk term

$$\mathcal{I}=\frac{i}{\hbar}\int\mathrm{d}^{d+1}x\left[\phi^{\mathrm{q}}\left(\ddot{\phi}^{\mathrm{cl}}-\bm{\nabla}^{2}\phi^{\mathrm{cl}}+V^{(1)}\right)+\sum_{n=1}^{+\infty}\frac{\left(\phi^{\mathrm{q}}\right)^{2n+1}}{2^{2n}(2n+1)!}V^{(2n+1)}\right]~{},$$

(6)

where $V^{(k)}\equiv\mathrm{d}^{k}V(\phi)/\mathrm{d}\phi^{k}|_{\phi\to\phi^{\mathrm{cl}}}$. The partition function can then be arranged in the form

$\displaystyle Z=\int{\mathcal{D}}\phi_{0}^{\mathrm{cl}}\,{\mathcal{D}}\pi_{0}^{\mathrm{cl}}\,W\big{(}\phi_{0}^{\mathrm{cl}},\pi_{0}^{\mathrm{cl}};t_{0}\big{)}\int\mathcal{D}^{\prime}\phi^{\mathrm{q}}\,\mathcal{D}^{\prime}\phi^{\mathrm{cl}}\;e^{-\mathcal{I}}~{},$

(7)

where

$\displaystyle W\big{(}\phi^{\mathrm{cl}}_{0},\pi^{\mathrm{cl}}_{0};t_{0}\big{)}\equiv\int{\mathcal{D}}\phi^{\mathrm{q}}_{0}\;\Big{\langle}\phi^{\mathrm{cl}}_{0}+\tfrac{\phi^{\mathrm{q}}_{0}}{2}\Big{|}\hat{\rho}(t_{0})\Big{|}\phi^{\mathrm{cl}}_{0}-\tfrac{\phi^{\mathrm{q}}_{0}}{2}\Big{\rangle}e^{-\frac{i}{\hbar}\int\mathrm{d}^{d}\mathbf{x}\;\phi^{\mathrm{q}}_{0}\pi^{\mathrm{cl}}_{0}}~{},$

(8)

is the initial Wigner function, in which the boundary term from Eq. (5) has fulfilled the role of the kernel in a Weyl transform of the initial density operator. Notice that we have relabelled $\pi_{0}^{\mathrm{cl}}\equiv\dot{\phi}_{0}^{\mathrm{cl}}$ to make contact with the Hamiltonian form of the path integral (see Ref. Hertzberg:2019wgx ). The prime on the integration measure indicates that the fields on the initial temporal boundary, such as $\phi_{0}(\mathbf{x})$ and $\pi_{0}(\mathbf{x})$, have been excluded.

Due to the Hermiticity of the density operator, i.e., $\hat{\rho}^{\dagger}=\hat{\rho}$, the initial Wigner function must be real-valued. In comparison, the bulk term $e^{-\mathcal{I}}$ is purely a phase term. Notice in addition that, to make the bulk path integral well-defined, the existence of $\ddot{\phi}^{\mathrm{cl}}$ requires two temporal boundary conditions, which are provided here by the Wigner function through $\phi_{0}^{\mathrm{cl}}$ and $\dot{\phi}_{0}^{\mathrm{cl}}$. This structure also has an impact on the critical point and related Lefschetz thimble as follows.

We first notice that the critical point³³3The critical point is actually a spacetime field configuration. We refer the reader to Refs. Mou:2019tck ; Mou:2019gyl for the concrete derivation of the saddle point. The subtlety lies at the turning point. of $\mathcal{I}$ satisfies

$\displaystyle\phi^{\mathrm{q}}=0\qquad\text{and}\qquad-\ddot{\phi}^{\mathrm{cl}}+{\bm{\nabla}}^{2}\phi^{\mathrm{cl}}-V^{(1)}=0~{},$

(9)

with the initial values of $\phi^{\mathrm{cl}}$ and $\dot{\phi}^{\mathrm{cl}}$ determined by the initial Wigner function $W(\phi_{0}^{\mathrm{cl}},\pi_{0}^{\mathrm{cl}};t_{0})$. That is, the critical point corresponds to a $\phi^{\mathrm{cl}}$ that follows the classical trajectory with initial data specified by $W(\phi_{0}^{\mathrm{cl}},\pi_{0}^{\mathrm{cl}};t_{0})$, and, as an initial value problem, there exists one and only one solution. This is the virtue of the two-step evaluation of the path integral. Namely, if we separate the path integral into the initial Wigner function $W(\phi_{0}^{\mathrm{cl}},\pi_{0}^{\mathrm{cl}};t_{0})$ and the dynamical part $e^{-\mathcal{I}}$, there will exist one and only one Lefschetz thimble for each initialization generated by the Wigner function. By Lefschetz thimble, we mean the manifold generated by the gradient flow originating from the critical point Witten:2010cx , and therefore the number of thimbles equals the number of critical points. For more on the two-step evaluation, we refer the reader to Refs. Mou:2019tck ; Mou:2019gyl .

The dynamical part $\mathcal{I}$ consists of odd terms in $\phi^{\mathrm{q}}$. If there are only linear terms in $\phi^{\mathrm{q}}$, we can integrate $\phi^{\mathrm{q}}$ out to obtain functional delta functions in $\phi^{\mathrm{cl}}$. Conversely, for nonlinear potentials that will yield odd terms in $\phi^{\mathrm{q}}$ of higher powers, imposing the same delta functions corresponds to the approximation of dropping these higher terms from Eq. (6). This is known as the Classical-Statistical (CS) approximation Moore:1996bn ; Aarts:1997kp ; Cooper:2001bd ; Berges:2006xc ; Epelbaum:2014yja . Note that this approximation maintains the non-linearity in $\phi^{\mathrm{cl}}$. In corollary, no such approximation is needed if there are only quadratic terms in the Lagrangian, and the CS approximation is then, in some sense, exact. The situation remains, however, non-trivial when the parameters vary in space and time. For instance, certain quantum effects in the early Universe can be well approximated via quadratic terms on the time varying background, and the ensemble average of classical evolutions will provide an honest description.

After integrating out $\phi^{\mathrm{q}}$, the CS approximation leads to

$\displaystyle Z=\int{\mathcal{D}}\phi_{0}^{\mathrm{cl}}\,\pi_{0}^{\mathrm{cl}}\,\mathcal{D}^{\prime}\phi^{\mathrm{cl}}\;W\big{(}\phi_{0}^{\mathrm{cl}},\pi_{0}^{\mathrm{cl}};t_{0}\big{)}\delta\Big{(}-\ddot{\phi}^{\mathrm{cl}}+\nabla^{2}\phi^{\mathrm{cl}}-V^{(1)}\Big{)}~{},$

(10)

where the delta function is understood in the functional sense. We see that the CS approximation to the partition function makes use of the critical points only, i.e., the classical trajectories, with their initializations distributed according to the initial Wigner function. Notice that the whole distribution function above is non-negative if the Wigner function is non-negative.

In comparison to the original expression (7), we may regard the $\phi^{\mathrm{q}}$ as hidden variables, only this time, the PDF is complex. However, this analogy is not quite complete, as we may also wish to measure $\phi^{\mathrm{q}}$-dependent operators. We want to stress that the complex PDF is a necessary condition for the violation of Bell inequalities, since if the distribution function is non-negative, one can always do the sampling and the generated samples cannot violate Bell inequalities. Now, with Eq. (10), we might speculate that there should not exist any violation of Bell inequalities in the free theory when a non-negative PDF about $\phi^{\mathrm{cl}}$ can be drawn. As we will see, this is not the case.

With regard the CS approximation, it will turn out that the temporal Bell-type inequalities due to Leggett and Garg Leggett:1985zz are of a richer structure, and we therefore choose to treat these before the more familiar spatial Bell inequalities.

The Leggett-Garg inequalities deal with measurements at different times. For the measurement operator, we choose $\hat{Q}={\rm sign}(\hat{\phi})$, which is a proper dynamical operator that maps the continuous variable $\phi$ into a dichotomous one Revzen ; Martin:2017zxs , taking values $\pm 1$.

It is useful to consider which correlators the experiment can measure and which two-point correlation functions they correspond to in the theory. In an experiment, we prepare an ensemble of sets, consistent with the same initial state $|\psi\rangle$. For each set, a measurement $Q_{1}\equiv\braket{\hat{Q}_{1}}=r$ is read out at $t_{1}$ and another measurement $Q_{2}\equiv\braket{\hat{Q}_{2}}=s$ is read out at a later time $t_{2}$, with $r,s=\pm$. The joint probability $P(r,s)$ can then be calculated Fritz , e.g., as

$\displaystyle P(+,+)=\frac{N(+,+)}{\sum_{r,s=\pm}N(r,s)}~{},$

(11)

where $N(r,s)$ is the number of sets of $Q_{1}=r$ and $Q_{2}=s$. Accordingly, the correlator is defined as

$\displaystyle C\equiv\sum_{r,s=\pm}rsP(r,s)~{}.$

(12)

For dichotomous variables, the correlator can further be related to the quantum two-point correlation function Fritz . To see this, recall that we have two probabilities for the two measurements:

$\displaystyle P(r)=|\langle r;t_{1}|\psi\rangle|^{2}\qquad\text{and}\qquad P(s|r)=|\langle s;t_{2}|r;t_{1}\rangle|^{2}~{},$

(13)

which correspond to the probability of finding $r$ at $t_{1}$, given the initial state $|\psi\rangle$, and the probability of finding $s$ at $t_{2}$, given the state $|r;t_{1}\rangle$, i.e., the state into which the first measurement collapses the system. The joint probability of finding $r$ at $t_{1}$ and $s$ at $t_{2}$ is then simply the product

$$P(r,s)=|\langle s;t_{2}|r;t_{1}\rangle|^{2}|\langle r;t_{1}|\psi\rangle|^{2}=\langle\psi|r;t_{1}\rangle\langle r;t_{1}|s;t_{2}\rangle\langle s;t_{2}|r;t_{1}\rangle\langle r;t_{1}|\psi\rangle~{}.$$

(14)

A dichotomous operator admits the following properties

$\displaystyle\hat{Q}$

$\displaystyle=\sum_{s=\pm 1}s|s\rangle\langle s|,\qquad\frac{1+s\hat{Q}}{2}=|s\rangle\langle s|~{},$

(15)

and we can therefore write

	$\displaystyle P(r,s)$	$\displaystyle=\langle\psi|\frac{1+r\hat{Q}_{1}}{2}\frac{1+s\hat{Q}_{2}}{2}\frac{1+r\hat{Q}_{1}}{2}|\psi\rangle$
		$\displaystyle=\frac{1}{8}\langle\psi|\left(2+2r\hat{Q}_{1}+s\hat{Q}_{2}+rs\{\hat{Q}_{1},\hat{Q}_{2}\}+s\hat{Q}_{1}\hat{Q}_{2}\hat{Q}_{1}\right)|\psi\rangle~{}.$		(16)

It is then straightforward to obtain the following equalities:

	$\displaystyle\sum_{r,s=\pm 1}P(r,s)$	$\displaystyle=1~{},$		(17a)
	$\displaystyle\sum_{r,s=\pm 1}rP(r,s)$	$\displaystyle=\langle\psi|\hat{Q}_{1}|\psi\rangle~{},$		(17b)
	$\displaystyle\sum_{r,s=\pm 1}sP(r,s)$	$\displaystyle=\frac{1}{2}\langle\psi|\hat{Q}_{2}+\hat{Q}_{1}\hat{Q}_{2}\hat{Q}_{1}|\psi\rangle~{},$		(17c)
	$\displaystyle\sum_{r,s=\pm 1}rsP(r,s)$	$\displaystyle=\frac{1}{2}\langle\psi|\{\hat{Q}_{1},\hat{Q}_{2}\}|\psi\rangle~{}.$		(17d)

The last of these [Eq. (17d)] in particular means that the correlator measured in the experiment via Eq. (12) is really the quantum two-point correlation function involving the anticommutator of the measurement operators. Interestingly, the above four equations indicate that the measurement procedure allows us to find $\langle\psi|\hat{Q}_{1}|\psi\rangle$ and $\langle\psi|\{\hat{Q}_{1},\hat{Q}_{2}\}|\psi\rangle$, but not $\langle\psi|\hat{Q}_{2}|\psi\rangle$. As we shall see, this is not the case in the CS approximation, where we are able to find $\langle\psi|\hat{Q}_{1}|\psi\rangle$ and $\langle\psi|\hat{Q}_{2}|\psi\rangle$, but not $\langle\psi|\{\hat{Q}_{1},\hat{Q}_{2}\}|\psi\rangle$.

An interesting observation is that the ${\rm Si}$ function appearing in Eq. (35) admits the following limit:

$\displaystyle\lim_{|G_{R}|\to 0}\frac{2}{\pi}\,{\rm Si}\left(\frac{2\tilde{\phi}^{\mathrm{cl}}_{2}\tilde{\phi}^{\mathrm{cl}}_{1}}{\hbar|G_{R}(t_{2},\mathbf{x}_{2};t_{1},\mathbf{x}_{1})|}\right)={\rm sign}(\tilde{\phi}^{\mathrm{cl}}_{2}){\rm sign}(\tilde{\phi}^{\mathrm{cl}}_{1})~{}.$

(39)

Since the retarded propagator $G_{R}(t_{2},\mathbf{x}_{2};t_{1},\mathbf{x}_{1})$ is causal, it vanishes when the separation of two points is spacelike, i.e., $(t_{2}-t_{1})^{2}<|\mathbf{x}_{2}-\mathbf{x}_{1}|^{2}$, which means that, for the free theory, $C^{\rm QFT}$ and $C^{\rm CS}$ give the same result in the case of spatial Bell inequalities. On the other hand, when the Wigner function is non-negative, the two-point functions in the form of Eq. (36) can never violate Bell inequalities. Thus, there can be no violation of Bell inequalities among spacelike correlation functions in free scalar field QFT, unless some entanglement exists in the initialization.

Beyond the free theory, we point out that it is a general property of QFT that $\phi^{\mathrm{q}}$ does not appear explicitly in the calculation of the spatial Bell inequalities. In fact, the following equations are valid in general in QFT:

	$\displaystyle C^{\rm QFT}(t_{2},\mathbf{x}_{2};t_{1},\mathbf{x}_{1})\big{|}_{\rm spacelike}$	$\displaystyle=\frac{1}{2}\Big{\langle}\{{\rm sign}(\hat{\phi}_{2}),\;{\rm sign}(\hat{\phi}_{1})\}\Big{\rangle}$
		$\displaystyle=\frac{\int{\mathcal{D}}\phi\;We^{-\mathcal{I}}\,\frac{1}{2}\left[{\rm sign}(\phi^{+}_{2}){\rm sign}(\phi^{+}_{1})+{\rm sign}(\phi^{-}_{1}){\rm sign}(\phi^{-}_{2})\right]}{\int{\mathcal{D}}\phi\;We^{-\mathcal{I}}}$
		$\displaystyle=\frac{\int{\mathcal{D}}\phi\;We^{-\mathcal{I}}\,{\rm sign}(\phi^{\mathrm{cl}}_{2}){\rm sign}(\phi^{\mathrm{cl}}_{1})}{\int{\mathcal{D}}\phi\;We^{-\mathcal{I}}}~{},$		(40)

with the reason discussed in Appendix B. In comparison, the CS approximation computes

$\displaystyle C^{\rm CS}(t_{2},\mathbf{x}_{2};t_{1},\mathbf{x}_{1})$

$\displaystyle=\frac{\int{\mathcal{D}}\phi\;We^{-\mathcal{I}^{\prime}}\,{\rm sign}(\phi^{\mathrm{cl}}_{2}){\rm sign}(\phi^{\mathrm{cl}}_{1})}{\int{\mathcal{D}}\phi\;We^{-\mathcal{I}^{\prime}}}~{},$

(41)

where $\mathcal{I}^{\prime}$ denotes the $\mathcal{I}$ with the higher-order terms in $\phi^{\mathrm{q}}$ discarded. In what follows, we will refer to those higher-order terms in $\phi^{\mathrm{q}}$ as “quantum vertices”, a name that can reflect their roles in Feynman diagrams Aarts:1997kp ; Epelbaum:2014yja ; Mou:2019tck . Since the CS approximation with Eq. (41) cannot violate any Bell-like inequalities, while the full quantum theory with Eq. (4) can, and given that the only difference between the two concerns the quantum vertices, we might speculate that these quantum vertices must have something to do with the origin of quantum entanglement. (Here, we equate quantum entanglement with the violation of Bell inequalities.) This seems to be the case with the following arguments.

We first notice that there exists a loophole in the above reasoning, which is related to the initialization. When quantum entanglement appears in the initial state, the Wigner function will be negative-valued in some field region. The negative distribution will make the standard sampling method difficult, if not impossible, to generate initializations for the CS approximation. That being said, if there exists some sophisticated re-sampling method for doing the initialization, the CS approximation will escape the constraint of not violating Bell inequalities.

On the other hand, the entanglement that already exists in the initialization does not help us to understand the origin of quantum entanglement. For this purpose, the best scenario is where the system starts from a non-negative Wigner function and gains some violation of spatial Bell inequalities during the evolution. Consider, for instance, the decay of a spinless particle into two photons. In this case, as we argued above, the CS, as an approximation, cannot capture such quantum entanglement. We can therefore conclude that it is those higher-order $\phi^{\mathrm{q}}$-terms in the action — the quantum vertices — that make quantum entanglement possible, and it is the absence of them that renders the CS approximation unable to capture quantum entanglement.

We further point out in the $\phi^{\mathrm{cl}}$-$\phi^{\mathrm{q}}$ representation that the quantum properties are usually accompanied by the appearance of $\phi^{\mathrm{q}}$. Recall, from the last section, the central role that the $\phi^{\mathrm{q}}$ plays in the Leggett-Garg or temporal Bell inequalities and here in the origin of the violation of spatial Bell inequalities, through the quantum vertices.

Another example is the commutation relation,

$\displaystyle\langle[\hat{\phi}(t,\mathbf{x}),\hat{\pi}(t,\mathbf{y})]\rangle=i\hbar\delta^{d}\left(\mathbf{x}-\mathbf{y}\right)~{},$

(42)

which is in fact a $\mathrm{q}$-$\mathrm{cl}$ two-point function. To see this, first recall, from the field-configuration representation of the path integral, that the field momentum can be computed via

$\displaystyle\pi(t,\mathbf{y})\equiv\lim_{\mathrm{d}t\to 0}\frac{\phi(t+\mathrm{d}t,\mathbf{y})-\phi(t,\mathbf{y})}{\mathrm{d}t}~{}.$

(43)

It is then straightforward to derive the commutation relation and obtain

	$\displaystyle\langle[\hat{\phi}(t,\mathbf{x}),\hat{\pi}(t,\mathbf{y})]\rangle=$	$\displaystyle\lim_{\mathrm{d}t\to 0}\frac{\langle\hat{\phi}(t,\mathbf{x})\hat{\phi}(t+\mathrm{d}t,\mathbf{y})-\hat{\phi}(t+\mathrm{d}t,\mathbf{y})\hat{\phi}(t,\mathbf{x})\rangle}{\mathrm{d}t}$
	$\displaystyle=$	$\displaystyle\lim_{\mathrm{d}t\to 0}\frac{\langle\phi^{-}(t,\mathbf{x})\phi^{-}(t+\mathrm{d}t,\mathbf{y})-\phi^{+}(t+\mathrm{d}t,\mathbf{y})\phi^{+}(t,\mathbf{x})\rangle}{\mathrm{d}t}$
	$\displaystyle=$	$\displaystyle-\lim_{\mathrm{d}t\to 0}\frac{\langle\phi^{\mathrm{q}}(t,\mathbf{x})\phi^{\mathrm{cl}}(t+\mathrm{d}t,\mathbf{y})\rangle}{\mathrm{d}t}$		(44a)
	$\displaystyle=$	$\displaystyle-\langle\phi^{\mathrm{q}}(t,\mathbf{x})\pi^{\mathrm{cl}}(t,\mathbf{y})\rangle~{}.$		(44b)

In the free theory, the relevant two-point function in Eq. (44a) is given by

$\displaystyle\langle\phi^{\mathrm{q}}(t_{1},\mathbf{x})\phi^{\mathrm{cl}}(t_{2},\mathbf{y})\rangle=-i\hbar\theta\left(t_{2}-t_{1}\right)\int\frac{\mathrm{d}^{d}\mathbf{p}}{(2\pi)^{d}}\frac{\sin\left(\omega_{\mathbf{p}}(t_{2}-t_{1})\right)}{\omega_{\mathbf{p}}}e^{i\mathbf{p}\cdot(\mathbf{x}-\mathbf{y})}~{},$

(45)

which, as a double-check, gives the correct delta function in the limit $\mathrm{d}t\to 0$.

In retrospect, when facing Eq. (37), the explicit Wigner function in the example, and noticing that $\phi_{0}^{\mathrm{cl}}$ and $\pi_{0}^{\mathrm{cl}}$ are independently distributed, one might doubt whether the initialization would respect the commutation relation. Now, as we see, one should in fact compute the two-point function in Eq. (44b) in order to check the commutation relation. Note that $\phi_{0}^{q}$ and $\pi_{0}^{\mathrm{cl}}$ appear in the kernel of the Weyl transform (8). Thus, by a substitution $\phi_{0}^{\mathrm{q}}W=\left(i\hbar\delta W/\delta\pi^{\mathrm{cl}}_{0}\right)$ and then integration by parts, we can verify that $\langle[\hat{\phi}(t_{0},\mathbf{x}),\hat{\pi}(t_{0},\mathbf{y})]\rangle=i\hbar\delta^{d}(\mathbf{x}-\mathbf{y})$. In particular, the validity does not depend on the detail of the Wigner function or the initial density matrix, but only on the Weyl transform.

We have studied QFT from the viewpoint of statistics. A general feature of the real-time path integrals is that the PDF is complex-valued, and, as a result, standard statistical tools, such as Markov chain or Monte Carlo, cannot be applied directly. This is reminiscent of the so-called “numerical sign problem”. To deal with the complex path integral, we can seek to use the Lefschetz thimble method, and we can summarize the following general properties of the real-time path integral (for further details, see Refs. Mou:2019tck ; Mou:2019gyl ):

• •

  The real-time path integral admits a two-part separation into the initial density matrix (via the Wigner function $W$) and the dynamical part ($e^{-\mathcal{I}}$). The initial Wigner function provides the initial data for the dynamical part, and it can be either non-negative or generally real-valued. The dynamical part, on the other hand, is purely a phase term, which is always situated on the edge of the convergent region in the complexified $\phi$-space.

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  Applying the Lefschetz thimble approach to the dynamical part, we find that the saddle points consist of classical trajectories. In particular, for each initialization, there exists a unique classical trajectory/saddle point/Lefschetz thimble.

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  The exponent in the dynamical part ($e^{-\mathcal{I}}$) includes only odd terms in $\phi^{\mathrm{q}}$. If we throw away any higher-order terms in $\phi^{\mathrm{q}}$, integrating out the $\phi^{\mathrm{q}}$ leads to functional delta functions that pick out the classical solutions. This leads to the CS approximation. Note that, after they have been integrated out, the $\phi^{\mathrm{q}}$ fields become hidden in the sense that we can no longer compute $\phi^{\mathrm{q}}$-dependent operators within the CS approximation.

In this paper, we have further pointed out that the complex PDF is a necessary condition for the violation of Bell-type inequalities, since, if the distribution function is non-negative, one can always apply the sampling method, and the so-generated samples cannot violate Bell-type inequalities. In this sense, the CS approximation is not expected to yield any violation of Bell-type inequalities, as it is restricted to non-negative PDFs.

This observation, however, leads to a conundrum that the free theory in a coherent state, for which the CS approach makes no approximations, admits a violation of the Leggett-Garg or temporal Bell inequalities. We resolved this puzzle by demonstrating that, for the example measurement operator $\hat{Q}={\rm sign}(\hat{\phi})$, the temporal two-point function depends on both $\phi^{\mathrm{cl}}$ and $\phi^{\mathrm{q}}$. Note that the PDF involving $\phi^{\mathrm{cl}}$ only is non-negative, but the PDF with $\phi^{\mathrm{q}}$ included is complex. This further validates our conclusion above that a complex PDF is needed for violations of Bell-type inequalities.