Manifest causality in quantum field theory with sources and detectors

We introduce a way to compute scattering amplitudes in quantum field theory including the effects of particle production and detection. Our amplitudes are manifestly causal, by which we mean that the source and detector are always linked by a connected chain of retarded propagators. We show how these amplitudes can be derived from a path integral, using the Schwinger-Keldysh"in-in"formalism. Focussing on phi-cubed theory, we confirm that our approach agrees with the standard S-matrix approach in the case of positive energy plane-wave scattering.


Introduction
Although relativistic quantum field theories are built with causality in mind, the way causality plays out at the level of the particle dynamics is not so clear. The usual "in-out" formalism places production and detection sources on the same footing and amplitudes involve only the Feynman propagator. Our approach replaces the usual expression for the scattering matrix by an "in-in" expectation value in which detection is distinct from production. It leads to manifestly causal results, in which the retarded progagator is prominent.
In section 2 we begin by presenting a general result for the expectation value of a hermitian operator that is some local function of field operators; we have in mind that the operator represents an observable. We do this in the presence of an external source and show that the effect of the source is transmitted to the detector via an unbroken chain of retarded propagators. Feynman propagators necessarily appear but never so as to break the causal link from source to detector.
In section 3 we present an expression for the calculation of scattering amplitudes. Since these can be expressed as expectation values of non-local products of field operators, we can use many of the results of the previous section. We then present the corresponding Feynman rules and illustrate their use in tree-level scattering.
In section 4 we show that our scattering amplitudes can be derived from a path integral in the "in-in" formalism. In section 5, we prove the equivalence of our scattering amplitudes, in the case of positive energy plane-wave scattering, with those obtained from the S-matrix. In appendix A, we describe the link with the cutting rules of thermal field theory.

Expectation values of local operators
We shall consider a single real scalar field, φ x ≡ φ(x), in the presence of some external disturbance that is active within a spacetime region R J . We suppose that a measurement taking place within a spacetime region R B is represented by a hermitian operator B that is some function of the field operator within the region, i.e. B ≡ B({φ x : x ∈ R B }), and that the disturbance is represented by a contribution to the Hamiltonian of the form Given a source whose strength is parametrised by γ, the sensitivity σ B of a detector making measurement B may be expressed as In what follows, we will show that σ B can be expressed using chains of retarded propagators from R J to R B , which implies that it vanishes outside of the forward light cone of the source.
In the interaction picture (which we always employ) the system evolves as where U (t, t A ) = 1 + (−i) is the evolution operator and Θ ijk... ≡ Θ(t i > t j > t k > . . .) is a Heaviside function, defined to be 1 if the time-ordered condition within the brackets is satisfied and 0 otherwise.
is the interaction Hamiltonian, including the effect of the source.
The expectation value of an operator B at a given time t 0 may then be written as Rather than express U (t, t A ) as a time-ordered exponential, we use a generalisation of the Baker-Hausdorff lemma [1], i.e. with B 0 ≡ B(t 0 ) we write with each term expressed as a set of nested commutators: Note that within each integral, t 0 > t 1 > . . . > t n > t A . To encompass the full extent of the influence of the source, t A must be a time prior to any point in R J . For the expectation value of B, we have where H int (t) = d 3 x H(x). Each term in the nth order perturbation operator F n involves n spacetime points x i , located progressively further back in time as i increases. An expression for the commutator of functions of field operators with the Pauli-Jordan function defined by [φ j , φ i ] = ∆ ij can be found in [2]. For our perturbation series, · · · ) and the sums k i run from zero to infinity, with the exclusion of the case in which all are simultaneously zero. Every one of the n iterations (r = 1, . . . , n) required to generate F n from F 0 generates a set of terms, each of which contains at least one factor of ∆ ir for some i < r. Each term in F n therefore refers to a set of spacetime points x 0 , x 1 , . . . , x i , . . . , x r , . . . , x n in which every member x i is connected to at least one earlier point x r by a retarded propagator. Thus every non-zero contribution to the sensitivity σ B of the detector must contain an unbroken chain of retarded propagators from a point in the source.
Phi-cubed theory. As an illustrative example we shall suppose that B(t 0 ) = φ(t 0 , x 0 ) = φ(x 0 ) and We assume that the system can be approximated by the vacuum of the non-interacting theory at t = −∞, i.e. we take |ψ A = |0 . The first-order commutator is where J i ≡ J(x i ). This gives is the retarded propagator for the free field. 1 Note the relative minus sign due to our definition of The second-order commutator F 2 = F 1 , H 2 is then Finally, the third-order commutator is We may now evaluate the expectation value of B. Any uncontracted field operators can be handled using Wick's Theorem and give rise to Feynman propagators: It then follows that the first term in the expectation value expansion eq. (2.7) is, from eq. (2.10), The second term, from eq. (2.11), involves an odd number of fields and therefore vanishes. The third term, from eq. (2.12) and depicted in figure 1, is Of particular note is the ∆ F 22 ∆ F 33 term on the second line (which corresponds to figure 1(h)). This includes an extra factor of 1/2 because there is a residual Heaviside function, Θ 23 , that cannot be absorbed into the retarded propagators. However, because of the symmetry under interchange of 2 ↔ 3 we can drop the time-ordering constraint in exchange for a symmetry factor of 1/2!. The same Heaviside function is also present in the term ∝ J 2 J 3 (see figure 1(l)) and its elimination also gives rise to a factor 1/2. For these two diagrams, we say that points 2 and 3 are 'equivalent'. Finally, the ∆ F 22 ∆ R 13 J 3 term on the final line (see figure 1(i)) also originally had a Θ 23 which we eliminated by combining it with a contribution ∝ ∆ F 33 ∆ R 12 J 2 . In this way all explicit time-ordering Heaviside functions disappear from the final expression.
From the first few nested commutators, it is evident that the majority of the terms in 0| B |0 are vacuum diagrams, which contribute nothing to the sensitivity of the detector to the source.
It is also apparent that there is a straightforward relationship between the set of diagrams that can be drawn and the form of a given nested commutator. The rules are listed below for the case B(t 0 ) = φ(t 0 ).

Feynman rules.
To compute F n we are to draw a set of skeleton graphs involving the n + 1 times from t 0 to t n . The graphs can be built up iteratively starting from the latest time, t 0 , and ending with the earliest time t n . We are to draw either a cubic interaction vertex (arising from gφ 3 /3!) or a source term (arising from γJ n φ) for all times t 1 to t n inclusive. For each time earlier than t 0 there must be at least one retarded propagator  Figure 2. The three contributions to F 2 . A horizontal row of dots represents a vertex and any uncontracted dots are understood as field operators. The graphs have an associated combinatoric and coupling factor: for (a) and (b) this is 18(g/3!) 2 and for (c) it is 6(g/3!)(−γJ 2 ).
heading forwards in time. This means that the vertices may include uncontracted field operators (i.e. there can be fewer than 3 retarded propagators at any vertex but never zero). For each vertex we associate a factor of g/3! and a factor of φ i for each uncontracted field operator (written in chronological order). For each instance of the source there is a factor of −γJ i and there is an overall factor of (−1) n . For each retarded propagator between x j (earlier) and x i (later) associate a factor of ∆ ij . There is a combinatoric factor for the number of different ways to contract the fields in forming the retarded propagators. This is the prescription to compute the operator F n and it is illustrated for the case of F 2 in figure 2.
To compute 0| F n |0 we take each skeleton, convert the ∆ ij to −∆ R ij and compute the vacuum Green's functions associated with the incomplete vertices using Wick's Theorem. There is a factor −i d 4 x i for all i > 0 and we must take care to absorb the time-ordering Heaviside functions into the retarded propagators: if any Heaviside functions remain then they can be eliminated provided we associate a symmetry factor of 1/m! if there are m 'equivalent' spacetime points (in the sense explained above). For example, there is a factor 1/2 for graphs (h) and (l) in figure 1.

Scattering amplitudes
We use the following expression for the scattering amplitude in the presence of a detector source K x and a production source J x : We will assume that K x has support only in the future of T and that the cubic interaction and source J x are turned off in that region. The situation is illustrated in figure 3. Equation (3.1) looks just like an "in-in" expectation value and as such it inherits many of the K J t T Figure 3. Source-to-detector scattering.
properties described in the previous section. Eq. (3.1) is a scattering amplitude in the sense that exp(i is rather like an S-matrix operator for determining the future of the state |T . For simplicity, we have assumed that the detector acts locally, e.g. there are no bilocal terms ∼ K(x, y)φ(x)φ(y).
In the case of n-to-two scattering, we can extract the relevant part of the amplitude, As in the previous section, and since we are assuming x 0 , y 0 > T , eq. (3.4) can be re-arranged by commuting φ x φ y through the time-evolution operator using the Baker-Hausdorff lemma, i.e. and To compute two-to-two scattering at tree-level we would need to compute the part of F 4 that is proportional to J 2 g 2 . The amplitude can also be obtained using the same Feynman rules as articulated in the previous section. The only difference is that the field operator that we are averaging can be a non-local polynomial of the field. This merely introduces extra points, all later than the time T , at which propagators may terminate. For each such point we will have a factor of −i d 4 Some examples at tree-level. We begin by considering the one-to-two amplitude, which is Notice that this includes a Feynman propagator coupling from the interaction vertex to the detector. There is however no violation of causality because the measurement is the coherent detection of two particles at points x and y, which is causal since one of those particles is constrained to lie in the future lightcone of the source by the unbroken chain of retarded propagators. In fact we should anticipate such superfically acausal correlations: they are a manifestation of entanglement.
We will now consider two-to-two scattering. In this case, the Feynman rules give The first line in the large braces of eq. (3.9) is the s-channel contribution, and the first set of terms in brackets on each of the third and fourth lines generates the t-channel contribution. The u-channel contribution is obtained from the [3 ↔ 4] interchange of the t-channel contribution. Note the residiual Θ 12 on the second line. It combines with a Θ 21 term after interchange (1 ↔ 2). Equation (3.9) can be simplified somewhat by symmetrizing the sources: Written in this form, there are only 5 distinct graphs to consider and these are illustrated in figure 4. We can make explicit contact with the corresponding S-matrix amplitude for the scattering of momentum eigenstates by promoting the sources K and J to operators in Fock space. Specifically, if we take K and J to be replaced by and we take the limit T → ∞ in such a way that the integrals over x 0 and y 0 can be approximated by integration over an infinite time domain, which allows the preparation of freely-propagating momentum eigenstates at t → ± ∞. We may then take the overlap of eq. (3.10) with two-particle "in" and "out" states, i.e.
After expressing the 5 propagators in eq. (3.10) in momentum space, the spacetime integrals can be performed leaving behind an overall energy-momentum conserving delta function. The final result is as expected: To illustrate the point, the last two graphs of figure 4, which correspond to the first two terms in the second line of eq. (3.10), contribute to the s-channel amplitude with weights 2 − 1 = 1.
The correspondence also works out for 2 → 3 at tree-level. In this case, there are three topologies to consider, as illustrated in figure 5(a). Figure 5(b) shows the graphs corresponding to the first topology in figure 5(a). Each graph should be summed over all allowed time orderings (i.e. subject to the rule that there must always be at least one retarded propagator heading forwards from any vertex) and agreement with the S-matrix calculation follows, e.g. for the graphs in figure 5(b) the relative weights are (in order from the first to the last graph) 3 − 1 − 1 − 1 − 6 + 2 + 2 + 2 + 1 = 1 and we have doubled the contribution from graphs 5 − 8 to account for the contribution where the Feynman and retarded propagators are swapped on the rightmost pair of outgoing legs. The two other types of graph shown in figure 5(a) also each have a total weight equal to 1. An all orders proof of the equivalence with the S-matrix, in the case of positive energy plane-wave scattering, is provided in section 5.

Path integral representation
In this section we will explain the connection of the operator-level amplitudes in the previous section to the retarded amplitudes of thermal field theory. To this end, we will apply the "in-in" or closed-time path (CTP) formalism due to Schwinger and Keldysh [3,4]. In real-time formulations of quantum field theory at finite temperature and density (see e.g. [5] and references therein), it is known that physical reaction rates must be calculated from the absorptive parts of retarded self-energies in order to obtain the correct quantum statistics [6,7]. We emphasise that retarded self-energies have arisen naturally in our treatment thus far (e.g. see the diagrams in figure 1).
The starting point of the "in-in" generating functional is the partition function: where the density operator ρ represents the detection subsystem, which we suppose to act in the time interval [T 1 , T 2 ]. Contained in this generating functional is the amplitude Γ JK from the previous section, since, in the Heisenberg picture, we can write Γ JK = 0|ρ|0 where To build the path integral, we prepare a set of N independent detectors, each described by a density operator ρ i and whose actions have exclusive support over infinitesimal intervals ∆t = t i+1 − t i = (T 1 − T 2 )/N , where t i ∈ [T 1 , T 2 ]. In the continuous limit, N → ∞, the density operator of the combined system of detectors may be written as the product integral In order to generate a path-integral representation of the partition function Z, we imagine perturbing the evolution of the system by means of some unphysical test source j x . Thus, we insert into eq. (4.1) unity in the form Notice that this insertion will generate two paths of evolution: C + , running forwards in time from T A to T B and C − , running backwards from T B and T A . It is the presence of these two anti-parallel integration contours that gives rise to the closed-time path C = C + ∪ C − of the "in-in" formalism. Hereafter, objects with time arguments confined to the positive (timeordered) branch are denoted by a subscript '+' and those with time arguments confined to the negative (anti-time-ordered) branch are denoted by a subscript '−'. We note that the time T A is the boundary time of the evolution of the system at which the initial conditions are specified. 2 Ultimately, we will take the limits T A → −∞ assuming that, asymptotically, the system is the free vacuum. By further inserting complete sets of eigenstates of the Heisenberg-picture field operator φ H x , we may develop the path-integral representation of the generating functional: where [dφ a z ] = T B T A [dφ a t (z)] dt denotes functional integration over '+' and '−' field configurations. The test sources and fields have been written in the doublet notation employed in [8][9][10], where and η ab = diag(1, −1). Hereafter, CTP indices, labelling the confinement of objects to the positive and negative branches of the CTP contour, are denoted by the lower-case Roman characters a, b = 1, 2 ≡ +, −. In the same notation, the action S[φ a ; T A , T B ] may be written where the interaction part is It contains the physical emission sources The tensor η abc , appearing in eq. (4.8), is defined such that (4.10) We may introduce an operator √ ρ and write the kernel of the density operator in the Again, by inserting complete sets of eigenstates of the Heisenberg field operator, we obtain where [dφ a z ] = T 2 T 1 [dφ a t (z)] dt and to simplify matters we henceforth assume T 1 = T B = T (this definition of T matches that in the previous section).
In general, the exponent K[φ a ; T 1 , T 2 ] will be expressed as an infinite series of convolutions of poly-local sources and fields of the form K abc...,xyz... φ a x φ b y φ c z · · · . However, for our purposes, we shall take K[φ a ; T 1 , T 2 ] to contain only a local source, i.e. where (4.14) Notice that K a x differs by a sign in the second element relative to the emission source J a x . This relative sign and the overall factor of 1/2 in eq. (4.13) arise from writing ρ = ( √ ρ) 2 .
Thus, we arrive at the form of the "in-in" generating functional for our choice of density operator: Completing the square in the free part of the action, we write  Figure 6. The two closed contours corresponding to the time integrals in eq. (4.18).
is the free 2 × 2 CTP matrix propagator. With this shift in the field, we may recast the "in-in" generating functional in the form where Z 0 [0] is the generating functional in the absence of interactions and for vanishing test sources j a, x and we have introduced the short-hand notation for functional derivatives with respect to the test sources. The two time integrals over the intervals [T A , T B ] and [T 1 , T 2 ] have given rise to two closed-time paths, as illustrated in figure 6, each corresponding to one of the subsystems into which the system has been partitioned. We will take T A → −∞, T B = T 1 = T and T 2 → +∞. By means of an orthogonal transformation, we may rotate to the Keldysh basis (e.g. see [5,11]): in which the elements of the CTP propagator comprise the retarded, advanced and Hadamard propagators (see appendix B). With this transformation where we have defined (4.23) Subsequently, contracting the CTP indices in the exponents of eq. (4.21), we obtain and we have used the fact that After suitable changes of variables and using the relations the Keldysh representation of the "in-in" generating functional eq. (4.24) may re-expressed as (4.28) Equation (4.28) is the main result of this section. Notice in particular that the physical source J couples only to the retarded propagator.
An example. By way of illustration, we now consider the specific case of the one-to-two amplitude, obtained by expanding the exponentials in eq. (4.28) to second order in the detection sources K and to first order in the emission sources J: Note that this amplitude is obtained directly from Z[j a ]. Acting with the rightmost functional derivative, we have Performing the remaining j + functional derivative, we may neglect the first term in the exponent, giving (4.31) Differentiating again and using the fact that ∆ R xx = 0, we obtain Finally, keeping only the connected diagrams, we have where integration variables have been relabelled for notational convenience. Performing the remaining functional derivatives, we arrive at the result which is in agreement with eq. (3.8) since x 0 , y 0 > T and z 0 < T , so the advanced contribution vanishes.

Relation to S-matrix
In this section, we will show that the amplitudes calculated in the preceding sections are equivalent to the corresponding S-matrix amplitudes for the scattering of positive energy plane-waves. We begin by considering a general set of tree-level graphs with N K outgoing external legs and N J incoming external legs. The number of vertices V and propagators P are given by V = N − 2 , P = 2N − 3 (5.1) in which N = N K + N J is the total number of external legs. Expanding each of the exponentials in eq. (4.28) to the appropriate order, we have where V = V + + V − and P = P + + P − . For convenience of notation, we have left implicit the fact that the j are set to zero externally. Using eq. (5.1), the factors of √ 2 in eq. (5.2) can be combined to give Performing the functional derivatives in the sources, we obtain Notice that the advanced contribution does not appear in the outgoing external legs since K x acts only in the future of all other vertices.
If all of the external four-momenta are on-shell the purely on-shell combination 2∆ F xy − ∆ R xy − ∆ A xy cannot occur in the internal lines of tree-level graphs by virtue of energymomentum conservation. As such, we set P − = N − K . Equation (5.4) then reduces to the following: This is to be compared with the corresponding term in the expansion of the usual formula for the S-matrix: As in eq. (5.6), we can affect the LSZ reduction to map from the vacuum amplitude to the S-matrix by promoting the external sources to operators in Fock space, i.e.
On contraction with the N K -particle "out" and N J -particle "in" Fock states, Γ N J →N K

JK
gives rise to a sum over all possible connected topologies. In the usual S-matrix approach, we would obtain a single graph for each topology. In the case of eq. (5.5) however, for each topology, we obtain a set of graphs with each graph contributing equally with a weight 2 1−N K . Comparing the remaining test sources and functional derivatives in eq. (5.5), the number of + type outgoing legs is given by Notice that, for φ 3 theory, N + K is always odd. Thus the set of graphs consistent with a given topology corresponds to the sum over all ways of drawing that topology with an odd number of + type outgoing legs. The number of ways of arranging N + K outgoing legs in a graph with a total of N K outgoing legs is the binomial coefficient: The total number of graphs consistent with a given topology is then obtained by summing over all odd 1 ≤ N + K ≤ N K , i.e.
This factor exactly cancels the overall factor of 2 1−N K in Γ JK and we are left with a sum over tree-level topologies, each with unit weight and entirely equivalent to the S-matrix result. This equivalence with the S-matrix can be extended beyond tree-level. Specifically, a general retarded Green's function (Γ n→m R ) can be obtained by summing over all circlings (see appendix A) except those of the m largest-time points, i.e.
and we have isolated the zero-circlings contribution corresponding to a graph built entirely from Feynman propagators. Since the scattering amplitude Γ n→m JK , derived in section 4, can be obtained from this retarded Green's function after convoluting with the source/detector functions, it follows that the same Γ n→m JK could be obtained using the corresponding Feynman Green function, Γ n→m F . This is because the second term on the right-hand side of eq. (5.11) vanishes if we impose that the incoming particles carry positive energy forwards in time, which is the case when evaluating the S-matrix. At tree-level, we have just shown that the combinatoric factor associated with the convolution of the Green's function and the sources is exactly as required for agreement with the S-matrix result, and this is sufficient to insure equivalence at all orders.
Note added: Whilst preparing the final version of this manuscript we became aware of reference [12], which presents a diagrammatic approach to the calculation of "in-in" expectation values similar to that presented in section 2 in the absence of external sources.

A Relation to unitarity cutting rules
We will now illustrate that the manifestly causal amplitudes derived from the operator and path-integral approaches are precisely the retarded amplitudes obtained by means of the Kobes-Semenoff unitarity cutting rules of the "in-in" formalism [13,14]. For the purposes of this section, we will omit to write the physical sources and convolution integrals associated with the external propagators and we take the limits T A → − ∞, so that interaction vertices are integrated over an infinite domain.
We begin by noting the following diagrammatic representation of the time-ordered (Feynman), anti-time-ordered (Dyson) and Wightman propagators (e.g. see [15]): We can follow the energy flow through a general graph built using these propagators since positive energy always flows from an uncircled into a circled vertex. The propagators satisfy Notice that, unlike the usual unitarity cutting rules applied to S-matrix theory, the Kobes-Semenoff cutting rules do not restrict diagrams to contain only positive energy flow. As we shall see below, negative energy flow is necessary for the construction of retarded diagrams and the restoration of manifest causality.
In terms of the propagators above, the retarded propagator can be expressed as In other words, circling plays no role for retarded propagators except to help keep track of the minus signs.
By virtue of the Kobes-Semenoff cutting rules, the one-loop, negative-frequency Wightman propagator is obtained by circling the right-most external vertex and summing over all possible circlings of the internal vertices. The one-loop retarded propagator then takes the form Since energy is conserved through the internal vertices the following circlings are identically zero: (A.5) Notice that the last diagram in eq. (A.4) is also vanishing. Adding to eq. (A.4) the vanishing diagrams from eq. (A.5), we make the following observation: the retarded diagram is obtained by summing over all possible circlings of vertices whilst leaving the outgoing, leftmost point uncircled [6]. is the truncated one-loop retarded self energy and denotes integration over the intermediate spacetime points z and z . It is interesting to see how this works starting from the truncated self-energy. In terms of the Kobes-Semenoff cutting rules, this is Following the same circling rules for the one-to-two scattering, i.e. we do not circle the "latest time" vertices that we anticipate coupling to detector sources K, the retarded contribution to the time-ordered 3-point function is given by which is in agreement with eq. (4.34) after convoluting with the source functions. We may repeat this diagrammatic manipulation for two-to-two scattering, which corresponds to 16 circlings:

B Propagator definitions
Here we collect together the definitions of the various propagators that appear.