For a free classical scalar field with the action defined in Equation (1), the classical stress-energy tensor is

which is equivalent to the tensor of Equation (4), but written in a slightly different form for convenience. When we make the transition to quantum field theory, we promote the field to a field operator . The fundamental problem of defining a quantum operator for the stress tensor is immediately visible; the field operator appears quadratically. Since is an operator-valued distribution, products at a single point are not well-defined. But if the product is point separated, , they are finite and well-defined.Let us first seek a point-separated extension of these classical quantities and then consider the quantum field operators. Point separation is symmetrically extended to products of covariant derivatives of the field according to

To carry out point separation on Equation (56), we first define the differential operator

from which we obtain the classical stress tensor as That the classical tensor field no longer appears as a product of scalar fields at a single point allows a smooth transition to the quantum tensor field. From the viewpoint of the stress tensor, the separation of points is an artificial construct, so when promoting the classical field to a quantum one, neither point should be favored. The product of field configurations is taken to be the symmetrized operator product, denoted by curly brackets: With this, the point separated stress-energy tensor operator is defined as While the classical stress tensor was defined at the coincidence limit , we cannot attach any physical meaning to the quantum stress tensor at one point until the issue of regularization is dealt with, which will happen in the next section. For now, we will maintain point separation so as to have a mathematically meaningful operator.The expectation value of the point-separated stress tensor can now be taken. This amounts to replacing the field operators by their expectation values, which is given by the Hadamard (or Schwinger) function

and the point-separated stress tensor is defined as where, since is a differential operator, it can be taken “outside” the expectation value. The expectation value of the point-separated quantum stress tensor for a free, massless () conformally-coupled () scalar field on a four-dimensional spacetime with scalar curvature is

We now turn our attention to the noise kernel introduced in Equation (12), which is the symmetrized product of the (mean subtracted) stress-tensor operator:

Since defined at one point can be ill-behaved, as it is generally divergent, one can question the soundness of these quantities. But as will be shown later, the noise kernel is finite for . All field-operator products present in the first expectation value that could be divergent, are canceled by similar products in the second term. We will replace each of the stress-tensor operators in the above expression for the noise kernel by their point-separated versions, effectively separating the two points into the four points . This will allow us to express the noise kernel in terms of a pair of differential operators acting on a combination of four and two-point functions. Wick’s theorem will allow the four-point functions to be re-expressed in terms of two-point functions. From this we see that all possible divergences for will cancel. When the coincidence limit is taken, divergences do occur. The above procedure will allow us to isolate the divergences and to obtain a finite result.Taking the point-separated quantities as more basic, one should replace each of the stress-tensor operators in the above with the corresponding point-separated version (60), with acting at and and acting at and . In this framework the noise kernel is defined as

where the four-point function is We assume that the pairs and are each within their respective Riemann normal-coordinate neighborhoods so as to avoid the problem that possible geodesic caustics might be present. When we later turn our attention to computing the limit , after issues of regularization are addressed, we will want to assume that all four points are within the same Riemann normal-coordinate neighborhood.Wick’s theorem, for the case of the free fields that we are considering, gives the simple product four-point function in terms of a sum of products of Wightman functions (we use the shorthand notation ):

Expanding out the anti-commutators in Equation (66) and applying Wick’s theorem, the four-point function becomes We can now easily see that the noise kernel defined via this function is indeed well-defined for the limit : From this we can see that the noise kernel is also well-defined for ; any divergence present in the first expectation value of Equation (66) has been cancelled by those present in the pair of Green’s functions in the second term, in agreement with the results of Section 3.

We will keep the points separated for a while so we can keep track of which covariant derivative acts on which arguments of which Wightman function. As an example (the complete calculation is quite long), consider the result of the first set of covariant derivative operators in the differential operator (57), from both and , acting on :

(Our notation is that acts at , at , at , and at .) Expanding out the differential operator above, we can determine which derivatives act on which Wightman function:If we now let and , the contribution to the noise kernel is (including the factor of present in the definition of the noise kernel): That this term can be written as the sum of a part involving and one involving is a general property of the entire noise kernel. It thus takes the form We will present the form of the functional shortly. First we note, that for and time-like separated, the above split of the noise kernel allows us to express it in terms of the Feynman (time ordered) Green’s function and the Dyson (anti-time ordered) Green’s function : This can be connected with the zeta-function approach to this problem [302] as follows: Recall that when the quantum stress-tensor fluctuations determined in the Euclidean section is analytically continued back to Lorentzian signature (), the time-ordered product results. On the other hand, if the continuation is , the anti-time ordered product results. With this in mind, the noise kernel is seen to be related to the quantum stress-tensor fluctuations derived via the effective action as The complete form of the functional is with

One of the most interesting and surprising results to come out of the investigations of the quantum stress tensor undertaken in the 1970s was the discovery of the trace anomaly [77, 102]. When the trace of the stress tensor is evaluated for a field configuration that satisfies the field equation (2), the trace is seen to vanish for massless conformally-coupled fields. When this analysis is carried over to the renormalized expectation value of the quantum stress tensor, the trace no longer vanishes. Wald [360] showed that this was due to the failure of the renormalized Hadamard function to be symmetric in and , implying that it does not necessarily satisfy the field equation (2) in the variable . (The definition of in the context of point separation will come next.)

With this in mind, we can now determine the noise associated with the trace. Taking the trace at both points and of the noise kernel functional (74) yields

For the massless conformal case, this reduces to which holds for any function . For as the Green’s function, it satisfies the field equation (2): We will assume only that the Green’s function satisfies the field equation in its first variable. Using the fact (because the covariant derivatives act at a different point than at which is supported), it follows that With these results, the noise kernel trace becomes which vanishes for the massless conformal case. We have thus shown, based solely on the definition of the point-separated noise kernel, that there is no noise associated with the trace anomaly. This result obtained in [305] is completely general since it is assumed that the Green’s function is only satisfying the field equations in its first variable; an alternative proof of this result was given in [258]. This condition holds not just for the classical field case, but also for the regularized quantum case, where one does not expect the Green’s function to satisfy the field equation in both variables. One can see this result from the simple observation used in Section 3; since the trace anomaly is known to be locally determined and quantum-state independent, whereas the noise present in the quantum field is non-local, it is hard to find a noise associated with it. This general result is in agreement with previous findings [58, 73, 206] derived from the Feynman–Vernon influence-functional formalism [107, 108] for some particular cases.

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