The stress-energy tensor in field theory

I came to physics research through general relativity, where the stress energy tensor plays a very important role, and where it has a single unambiguous meaning as the functional derivative of the theory with respect to metric perturbations. In flat-space quantum field theory, some texts present the stress tensor this way, while some present the stress tensor as a set of Noether currents associated with spatial translations. These definitions are usually presented as being equivalent, or rather, equivalent up to the addition of some total derivative that doesn't affect the physics.

However, this is not actually the case. The two stress tensors differ by terms that can be made to vanish classically, but that have an important effect in the quantum theory. In particular, the Ward identities of the two different stress tensors are different. This has caused me a lot of grief over the years, as I've tried to compare equations between texts that use two different definitions of the stress tensor, and been unable to reconcile the differences. This is compounded by the fact that even in cases where the stress tensor is defined in terms of a functional derivative, it is occasionally defined not by the equation that is usually written
$$T_{\mu \nu} = - \frac{2}{\sqrt{|g|}} \frac{\delta S}{\delta g^{\mu \nu}},$$
which indicates a functional derivative of the action with respect to an independent metric variation, but rather by a functional derivative with respect to a family of perturbations that act simultaneously on the metric and on the fields, with the coupling being chosen to give the fields some "dimension."

I've found all of these various ideas very confusing to try to understand, so I'm putting this note together to try to discuss, unambiguously, the various objects that people call "stress tensors" in classical and quantum field theory and how they differ from one another.

In section 1, I take up the issue of what it means for a diffeomorphism to act on a field. I review push-forwards of tensor fields under diffeomorphisms, Lie derivatives of tensor fields, and the linear transformations that are sometimes used to supplement push-forwards in theories where doing so gives a symmetry of the Lagrangian.

Section 2 is dedicated to what I will call a functional stress tensor $T_{ab},$ which is a stress tensor defined by a functional derivative with respect to metric perturbations. I give its definition in both classical and quantum field theories, and in the quantum theory discuss its Ward identities. I also discuss the case of "dimensionful" fields, where the functional derivative is supplemented by some dependence of field configurations on the metric.

In section 3, I discuss the current stress tensor $T_{ab}^{C},$ which is defined for field theories in flat space as the Noether current of translations. I discuss inequivalences between $T_{ab}^{C}$ and $T_{ab}$ for quantum field theories.

NOTE: While I was finalizing these notes, a paper appeared on the arXiv that I think deals with some of the same ambiguities that I've tried to handle systematically in this post. In particular, they discuss systematic ways of adding "non-divergence improvement terms" to the current stress tensor to make it match the functional stress tensor; I explain what this means in section 3. I haven't read the paper closely, but it looks like their method involves classifying a large set of ambiguities in the definition of the current stress tensor by generalizing the derivation of Noether's theorem to allow for higher-derivative couplings to the smearing field $\rho(x).$

Prerequisites: Abstract index notation, as used in general relativity. Differential forms at the level of appendix of Wald's book on general relativity. Path integral formulation of quantum field theory. Lagrangian field theory at the level of my post on Ward identities.

Table of Contents

1. How do diffeomorphisms act on fields?
2. Functional stress tensors in general backgrounds
3. The current stress tensor
   

1. How do diffeomorphisms act on fields?

(NOTE: Readers comfortable with push-forwards and Lie derivatives on general manifolds can skip to the last two paragraphs of this section.)

 

For the purposes of this post, all fields in a theory will be tensor fields, i.e., sections of a tensor bundle over a base manifold $\mathcal{M}.$ Now, let $\psi : \mathcal{M} \rightarrow \mathcal{M}$ be a diffeomorphism of the manifold. There is a natural way that $\psi$ acts on tensor fields, called the push-forward or pull-back. The idea is very simple.

A function $f : \mathcal{M} \rightarrow \mathbb{R}$ can be pulled back from a point $p$ to a point $\psi^{-1}(p)$ as
$$(\psi_* f)|_{\psi^{-1}(p)} = f|_{p}.$$

The crucial feature of a pullback is that the function $\psi_* f$ at the point $\psi^{-1}(p)$ depends only on the value of $f$ at $\psi(p).$ For functions, this relationship is rather obvious; we could equivalently write $\psi_* f = f \circ \psi.$ But we can extend this idea to more general tensors. If $v^{a}$ is a vector field, then we define the push-forward of $v^{a}$ at the point $\psi(p)$ by requiring that for all functions $f$ smooth in a neighborhood of $\psi(p)$, the push-forward $(\psi^* v)^a|_{\psi(p)}$ satisfies
$$((\psi^* v)^{a} \nabla_{a} f)|_{\psi(p)} = (v^{a} \nabla_{a} (\psi_* f))|_{p}.$$

Since tensors of the form $\nabla_a f$ span the space of dual vectors, this identity completely specifies $\psi^* v$ at $\psi(p).$

We can play a similar game to define the pullback of a dual vector $\omega_a$ by
$$((\psi_* \omega)_{a} v^a)|_{\psi^{-1}(p)} = (\omega_a (\psi^* v)^a)|_{p}.$$

We can iterate this kind of definition to define the push-forward of an arbitrary number of up-indices, and the pull-back of an arbitrary number of down indices. But because $\psi$ is a diffeomorphism, we can define a push-forward of down-indices by pulling back with $\psi^{-1}.$ In this way, the push-forward of a general tensor $S^{a_1 \dots a_m}{}_{b_1 \dots b_n}$ is defined by
$$\begin{split}\left((\psi^* S)^{a_1 \dots a_m}{}_{b_1 \dots b_n} (\omega_{1})_{a_1} \dots (\omega_{m})_{a_m} (v_1)^{b_1} \dots (v_n)^{b_n}\right)|_{\psi(p)} \\ = \left( S^{a_1 \dots a_m}{}_{b_1 \dots b_n} (\psi_* \omega_{1})_{a_1} \dots (\psi_* \omega_n)_{a_n} (\psi^{-1, *} v_1)^{b_1} \dots (\psi^{-1, *} v_n)^{b_n} \right)|_{p}.\end{split}$$

We can define a general pullback similarly.

 

We can use these ideas to define the Lie derivative of a tensor field. In order to define a derivative of a tensor field $S$ (whose indices I have suppressed), we need to be able to compare the value of $S$ at the point $p$ to its value at a nearby point $q$. But there is no way to do this canonically; tensors at different points in a manifold live in different vector spaces that cannot be identified. One common way to do this comparison is with a covariant derivative $\nabla_a,$ usually defined with respect to a metric, which lets us compare the value of $S$ at $p$ to the value it would have at $p$ if it were parallelly transported from $q.$ The Lie derivative gives us a different notion of the derivative of a tensor field, comparing tensor fields at different points using push-forwards/pullbacks rather than parallel transport.

Let $p$ be a point in a manifold, $S$ a tensor field in a neighborhood of that point, and $\xi^{a}$ a smooth vector field in that neighborhood. Integrating $\xi^{a}$ up to parameter $\epsilon$ gives a diffeomorphism $\psi^{\epsilon}.$ The Lie derivative of $S$ with respect to $\xi$ is defined by

$$\mathscr{L}_{\xi} S = \lim_{\epsilon \rightarrow 0} \frac{\psi^{\epsilon}_* S|_{p} - S|_{p}}{\epsilon}.$$

Heuristically, the Lie derivative says: "look a bit ahead to the point $p + \epsilon \xi,$ take $S$ at that point, pull it back to $p,$ and compare what you get to the actual value of $S$ at $p.$" The Lie derivative will be essential in the next section.

Now, since push-forwards and pullbacks are equivalent under the substitution $\psi \mapsto \psi^{-1},$ we will henceforth talk only about push-forwards. We have shown that push-forwards provide a natural action of diffeomorphisms on tensor fields But this certainly isn't the only way for diffeomorphisms to act! More generally, for example, we could push-forward a tensor field by the diffeomorphism $\psi,$ then act at each point with a linear transformation $A_{\psi}.$

Allowing this more general notion of the action of a diffeomorphism is essential for finding symmetries of certain field theories. For example, in $D$-dimensional Euclidean space, the free scalar theory with action
$$\int d^{D} x \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi$$

is not invariant under push-forwards of $\phi$ by scale transformations $\psi(x) = \lambda x$ except in $D=2$; however, it is invariant if we also act on $\phi$ at each point by the linear transformation $\phi \mapsto \lambda^{- (D-2)/2} \phi.$ So while the push-forward scale transformation is not a symmetry of the theory, the push-forward-plus-linear scale transformation is. In the next section, we'll put this point aside and focus solely on infinitesimal trasnformations of fields induced by push-forwards, but we'll take it up again in section 2.3 when we discuss how some non-symmetric local push-forwards can be turned into infinitesimal symmetries by composing them with local transformations of the fields.

2. Functional stress tensors in general backgrounds

Suppose we have a field theory on a manifold $\mathcal{M}$ with metric $g_{ab}$ and Lagrangian $\mathbf{\mathcal{L}}$. As explained in section 1 of my post on Ward identities, the Lagrangian is best thought of as an $n$-form, with $n = \operatorname{dim}(\mathcal{M}).$ The Lagrangian depends on a collection of fields $\phi^{\alpha},$ which are sections of a vector bundle over $\mathcal{M}.$ This data is enough to define a classical field theory; to define a quantum field theory, we must also specify a path integral measure $\mathcal{D} \phi^{\alpha}.$

In the first subsection, I will define a canonical functional stress tensor in classical field theories and show how it controls how the action responds to local diffeomorphisms. In the second subsection, I will define the canonical functional stress tensor in quantum field theories and discuss its Ward identities. In the final subsection, I will expand my definition of "functional stress tensor" to include functional derivatives with respect to coupled variations between matter fields and the metric.

2.1 Classical field theory

As explained in section 1 of my post on Ward identities, under any infinitesimal transformation $\phi^{\alpha}(x) \mapsto \phi^{\alpha}(x) + \lambda \delta \phi^{\alpha}(x),$ the Lagrangian changes as
$$\delta \mathcal{L} = \mathbf{E}(\phi)_{\alpha} \delta \phi^{\alpha} + d \boldsymbol{\theta}.$$

In this expression, $\mathbf{E}$ is an $n$-form that depends only on the background field configuration, called the equation of motion of the theory. The $(n-1)$-form $\boldsymbol{\theta}$ is a functional of both $\phi$ and $\delta \phi$, and is ambiguous up to the addition of an exact form $\boldsymbol{\theta} \mapsto \boldsymbol{\theta} + d \boldsymbol{\alpha}.$

There is a fairly special kind of infinitesimal transformation in field theory, which is one induced by pushing forward all fields in the theory by a local diffeomorphism. If $\xi^a$ is a smooth vector field in a neighborhood of the point $x,$ then the corresponding local push-forward is given by

$$\phi^{\alpha}(x) \mapsto \phi^{\alpha}(x) - \lambda (\mathscr{L}_{\xi} \phi)^{\alpha}(x).$$

(The minus sign here might seem out of place; after all, aren't we pushing the field forward? Remember that because we are pushing the field forward, the new value of $\phi^{\alpha}$ at $x$ ought to be coming from the old value of $\phi^{\alpha}$ at $x - \lambda \xi.$)

There is a very useful tool for understanding how an infinitesimal push-forward acts on the Lagrangian $\mathbf{\mathcal{L}},$ which is to ask first, "how would this transformation act if I also let it affect the metric?" Remember, the Lagrangian is a local function of the metric as well as the fields $\phi^{\alpha},$ so we could easily let a field transformation act on the metric if we wanted to. If we were to allow this, then a general perturbation $\phi \mapsto \phi + \lambda\delta \phi, g \mapsto g + \lambda \delta g,$ would change the Lagrangian as
$$\delta_{\phi, g} \mathbf{\mathcal{L}} = \mathbf{E}^{(\phi)}_{\alpha} \delta \phi^{\alpha} + \mathbf{E}^{(g)}_{ab} \delta g^{a b} + d \boldsymbol{\theta}^{(\phi)} + d \boldsymbol{\theta}^{(g)}. \tag{1}$$

The terms split in such a nice way — equations of motion and $\boldsymbol{\theta}$ terms written for $\phi$ and $g$ separately — because this expression must be linear in the perturbation, so there can be no terms containing both $\delta \phi$ and $\delta g.$

 

If we were to transform by a local push-forward, sending $\phi$ to $\phi - \lambda \mathscr{L}_{\xi} \phi$ and $g$ to $g - \lambda \mathscr{L}_{\xi} g,$ then the change in the Lagrangian would be

$$\delta_{\phi, g} \mathbf{\mathcal{L}} = - \mathscr{L}_{\xi} \mathbf{\mathcal{L}}.$$

At this point, we can apply Cartan's wonderful formula for Lie derivatives of differential forms to obtain
$$\mathscr{L}_{\xi} \mathbf{\mathcal{L}} = \xi \cdot d \mathbf{\mathcal{L}} + d(\xi \cdot \mathbf{\mathcal{L}}).$$

Because $\mathbf{\mathcal{L}}$ is a top-level form, its exterior derivative vanishes, and we are left with

$$\delta_{\phi, g} \mathbf{\mathcal{L}} = - d(\xi \cdot \mathbf{\mathcal{L}}). \tag{2}$$

Note that the Lagrangian changes by an exterior derivative; this is expected, because pushing forward all fields including the metric by a local diffeomorphism is a symmetry of any diffeomorphism-invariant theory.

The reason we did all of this is because it gives us an extraordinarily elegant formula for the transformation of the Lagrangian under a push-forward that only affects $\phi^{\alpha}.$ Setting equations (1) and (2) equal to each other, we obtain

$$\mathbf{E}^{(\phi)}_{\alpha} (\mathscr{L}_{\xi} \phi)^{\alpha} = - d(\xi \cdot \mathbf{\mathcal{L}}) - \mathbf{E}^{(g)}_{ab} (\mathscr{L}_{\xi} g)^{a b} - d \boldsymbol{\theta}^{(\phi)}_{\xi} - d \boldsymbol{\theta}^{(g)}_{\xi}, \tag{3}$$

where we have added subscripts $\xi$ to the $\boldsymbol{\theta}$ forms to remind ourselves that they depend on our particular choice of variation.

The reason equation (3) is so beautiful is that many of its terms are total derivatives! The only lingering term, $- \mathbf{E}_{ab}^{(g)} (\mathscr{L}_{\xi} g)^{ab},$ is so special that we will take the rest of this section to study it in detail. Because $\mathbf{E}_{ab}^{(g)}$ is a top-level form, it must be proportional to the volume form, and so we can write
$$\mathbf{E}_{ab}^{(g)} = - \frac{1}{2} T_{ab} \boldsymbol{\epsilon}.$$

The tensor $T_{ab}$ is called the stress-energy tensor of the theory. It is ambiguous up to addition of an antisymmetric term, since the only way it ever shows up in our equations is via contraction into the symmetric tensor $(\mathscr{L}_{\xi} g)^{ab}$; to resolve this ambiguity, we will always choose $T_{ab}$ to be symmetric. The factor of $1/2$ is a convention which makes our final equations simpler. As a brief comment, recalling that in any system of coordinates $\boldsymbol{\epsilon}$ is proportional to $\sqrt{|g|}$, the expression
$$\delta_{g} \mathbf{\mathcal{L}} = - \frac{1}{2} T_{ab} \delta g^{ab} \boldsymbol{\epsilon}$$

is consistent with the coordinate-dependent notation that appears in most sources,

$$T_{\mu \nu} \equiv - \frac{2}{\sqrt{|g|}} \frac{\delta S}{\delta g^{\mu \nu}},$$

where $S = \int_{\mathcal{M}} \mathbf{\mathcal{L}}$ is the action.

(You'll sometimes also see the expression
$$T^{\mu \nu} \equiv \frac{2}{\sqrt{|g|}} \frac{\delta (L \sqrt{|g|})}{\delta g_{\mu \nu}}.$$

These expressions are equivalent; taking the functional derivative with respect to $g$ instead of $g^{-1}$ introduces a relative minus sign.)

We can now put equation (3) in a slightly more instructive form by writing $(\mathscr{L}_{\xi} g)^{ab}$ explicitly as $- \nabla^{a} \xi^b - \nabla^{b} \xi^{a},$ which gives the expression

$$\mathbf{E}^{(\phi)}_{\alpha} (\mathscr{L}_{\xi} \phi)^{\alpha} = - T_{ab} \boldsymbol{\epsilon} \nabla^{a} \xi^{b} - d\left(\boldsymbol{\theta}^{(\phi)}_{\xi} + \boldsymbol{\theta}^{(g)}_{\xi} + \xi \cdot \mathbf{\mathcal{L}}\right).$$

If $\xi^a$ is compactly supported, or if it is sufficiently well behaved at infinity, then integrating the $\nabla_a$ derivative by parts gives

$$\boldsymbol{\epsilon} T_{ab} \nabla^{a} \xi^{b} = \boldsymbol{\epsilon} \nabla^{a} (\xi^b T_{ab}) - \boldsymbol{\epsilon} \xi^{b} \nabla^{a} T_{ab}.$$

But $\boldsymbol{\epsilon} \nabla^{a} v_a$ for any vector $v_a$ is equivalent to $d(v \cdot \boldsymbol{\epsilon}),$ i.e., it is an exact form. (This is discussed toward the end of section 1 of my post on Ward identities.) Putting all this together, we obtain the identity

$$\mathbf{E}^{(\phi)}_{\alpha} (\mathscr{L}_{\xi} \phi)^{\alpha} = \xi^{b} \nabla^{a} T_{ab} \boldsymbol{\epsilon} - d\left(\boldsymbol{\theta}^{(\phi)}_{\xi} + \boldsymbol{\theta}^{(g)}_{\xi} + \xi \cdot \mathbf{\mathcal{L}} + (\xi \cdot T) \cdot \boldsymbol{\epsilon} \right) \tag{4}.$$

 

Now, we can write the change in the action when $\phi$ is replaced by $\phi - \lambda \mathscr{L}_{\xi} \phi$ and $g$ is kept fixed as

$$\delta_{\xi} S = \int_{\mathcal{M}} \left[- \mathbf{E}^{(\phi)}_{\alpha} (\mathscr{L}_{\xi} \phi)^{\alpha} + d \boldsymbol{\theta}^{(\phi)}_{\xi} \right].$$

If we plug in equation (4) and drop surface terms, then we obtain the identity

$$\delta_{\xi} S = - \int_{\mathcal{M}} \xi^b \nabla^a T_{ab} \boldsymbol{\epsilon}. \tag{5}$$

This is a beautiful expression! It gives us a very simple formula for how the action of a field theory changes under a local push-forward of the non-metric fields, written only in terms of a single tensor $T_{ab}.$

 

Furthermore, because $\xi^{a}$ is arbitrary (at least, so long as it doesn't have violent behavior at infinity that stops the surface terms from vanishing), equation (5) implies that $\nabla^{a} T_{ab}$ vanishes on field configurations satisfying the equations of motion. This is because on field configurations satisfying the equation of motion, the action must be stationary against arbitrary infinitesimal transformations, which means that equation (5) must vanish; this is the statement that the stress-energy tensor is conserved "on-shell," i.e., on field configurations satisfying the equations of motion.

 

NOTE: The symmetry of the stress tensor was essential in proving that it is conserved on configurations satisfying the equation of motion. Symmetrizing the stress tensor let us write $T_{ab} (\nabla^{a} \xi^b + \nabla^b \xi^a)$ as $2 T_{ab} \nabla^{a} \xi^{b}$; if we hadn't symmetrized the stress tensor first, equation (4) would be different, and we would have no guarantee that it was conserved. This is good, of course, because there are plenty of non-conserved antisymmetric tensors, so it shouldn't be the case that any possible stress tensor $T_{ab} + H_{ab}$ with $H$ antisymmetric is conserved.

 

2.2 Quantum field theory

Quantum field theories are determined by the path integral weight, which is the product of the path integral measure and the exponentiated action:

$$W = \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]},$$

where $\gamma$ is $i$ in Lorentz signature and $(-1)$ in Riemannian signature. The path integral weight depends both on the matter fields $\phi^{\alpha}$ and on the metric $g_{ab}.$ We will assume that the dependence of $W$ on the metric is given by a local integral; that is, under the infinitesimal perturbation $g_{ab} \mapsto g_{ab} + \lambda \delta g_{ab},$ we have

$$\delta W = - \frac{\gamma}{2} \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int_{\mathcal{M}} T_{ab} \delta g^{ab} \boldsymbol{\epsilon}. \tag{6}$$

In the previous subsection, we were able to derive an equation like this for classical field theories, since we know the change in the action is proportional to the integral of $\mathbf{E}^{(g)}_{ab} \delta g^{ab}.$ In quantum field theory, the existence of an equation like this one is an additional assumption we must place on the theory, since we cannot prove that the path integral measure also transforms this way. It seems eminently reasonable, though — the path integral measure is supposed to be local in the fields, so what else could we possibly write down? The only additional term I can think of would be a surface term, i.e. a total derivative

$$\int_{\mathcal{M}} \nabla_{a} (P^{a}{}_{b c} \delta g^{bc}) \boldsymbol{\epsilon}.$$

We can't rule these out a priori, so we should only really trust equation (6) for metric perturbations that are sufficiently well behaved at infinity. To actually determine what falloff conditions must be imposed on $\delta g^{ab}$ for a putative surface term to vanish, we could proceed by dimensional analysis — $P^{a}{}_{bc}$ ought to be constructed locally from the fields and their derivatives, so one could estimate its slowest possible falloff at infinity from its index structure and from the falloff conditions imposed on the field configurations, and from this determine what falloff conditions must be imposed on metric perturbations. For most of the rest of this post, we'll be considering compactly supported metric perturbations, so the issue of surface terms won't arise at all; I mention it only because I think there are some cases where it might actually be important to think about.

If there exists a symmetric tensor satisfying equation (6), it is called the quantum stress tensor of the theory. It is important to note that in general, the tensor $T_{ab}$ defined by equation (6) will be different from the classical stress tensor of the theory. The two expressions agree only if the path integral measure is completely insensitive to arbitrary compact metric perturbations.

Now we use the same trick as in the previous subsection, which is to ask: how would $W$ change if I made the simultaneous substitution $g \mapsto g - \lambda \mathscr{L}_{\xi} g,$ $\phi \mapsto \phi - \lambda\mathscr{L}_{\xi} \phi$? Let's fix $\xi^{a}$ to be compactly supported for concreteness. If our quantum field theory is diffeomorphism covariant — as it certainly should be, unless we want the laws of physics to depend on the coordinate system we use to write them down — then the path integral weight must be unchanged under this substitution:

$$\delta_{\xi} \left(\mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]}\right) = 0.$$

So if we leave $g$ fixed and only vary $\phi$ by a local diffeomorphism, then the path integral weight should change by

$$\delta_{\xi, \phi} \left(\mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]}\right) = - \delta_{\xi, g} \left(\mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]}\right) =  \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int_{\mathcal{M}} T_{ab} \nabla^{a} \xi^{b} \boldsymbol{\epsilon}. \tag{7}$$

This is the fundamental equation controlling how the quantum field theory responds to local push-forwards of the fields. Since $\xi^a$ is compactly supported, we can integrate by parts to obtain

$$\delta_{\xi, \phi} \left(\mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]}\right) = - \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int_{\mathcal{M}} \xi^b \nabla^a T_{ab} \boldsymbol{\epsilon}.$$

Finally, we give the Ward identity that explains how the stress tensor behaves within correlation functions. The correlation function

$$\langle \phi^{\alpha_1}(x_1) \dots \phi^{\alpha}_n (x_n) \rangle$$

can be written after a change of variables in the path integral as

$$\frac{1}{Z} \int \mathcal{D}[\phi^{\alpha} - \lambda (\mathscr{L}_{\xi} \phi)^{\alpha}] e^{\gamma S[\phi - \lambda \mathscr{L}_{\xi} \phi]} (\phi^{\alpha_1}(x_1) - \lambda (\mathscr{L}_{\xi} \phi)^{\alpha}) \dots.$$

The linear-in-$\lambda$ terms are, collectively,

$$\gamma \langle \int_{\mathcal{M}} T_{ab}(x) \nabla^{a} \xi^{b}(x) \boldsymbol{\epsilon}(x) \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle - \sum_{j=1}^{n} \mathscr{L}_{\xi, j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n)\rangle.$$

Since the linear terms must sum to zero, we obtain the integrated Ward identity

$$\gamma \langle \int_{\mathcal{M}} T_{ab}(x) \nabla^{a} \xi^{b}(x) \boldsymbol{\epsilon}(x) \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle = \sum_{j=1}^{n} \mathscr{L}_{\xi, j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n)\rangle.$$

Or, if you prefer,

$$\gamma \langle \int_{\mathcal{M}} \xi^b(x) \nabla^a T_{ab}(x) \boldsymbol{\epsilon}(x) \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle = - \sum_{j=1}^{n} \mathscr{L}_{\xi, j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n)\rangle \tag{8}.$$

 In particular, if $\xi$ is supported away from the insertion points $x_1, \dots, x_n,$ then the right-hand side of this equation vanishes, which is the statement that correlation functions including $\nabla^{a} T_{ab}(x)$ vanish unless other operators are inserted at the point $x.$

 

The Ward identity written in equation (8) can be recast in the usual "distributional" form by noting that the Lie derivative of a tensor $(\mathscr{L}_{\xi} \phi)^{\alpha}$ can always be related to its covariant derivative by an expression like

$$(\mathscr{L}_{\xi} \phi)^{\alpha} = \xi^{b} \nabla_{b} \phi^{\alpha} + M_{ab}^{\alpha} \nabla^{a} \xi^{b},$$

where $M^{\alpha}_{ab}$ is linear in the field. (This can be seen, for example, from the general formula for a Lie derivative given in equation C.2.14 of Wald's book.) In particular, this implies that if $\rho(x)$ is a smooth function, then we have

$$(\mathscr{L}_{\rho \xi} \phi)^{\alpha} = \rho (\mathscr{L}_{\xi} \phi)^{\alpha} + M_{ab}^{\alpha} \xi^b \nabla^a \rho.$$

So if, say, we let $\rho$ be a function supported in a neighborhood of $x_1$ that does not intersect any other point $x_j$, then we may write

$$\begin{split}& \gamma \int_{\mathcal{M}} \rho(x) \langle  \xi^b(x) \nabla^a T_{ab}(x) \boldsymbol{\epsilon}(x) \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle \\ & = - \rho(x_1) \langle (\mathscr{L}_{\xi}\phi^{\alpha_1}(x_1)) \dots \phi^{\alpha_n}(x_n)\rangle - (\nabla^{a} \rho(x_1)) \xi^b(x_1) \langle M^{\alpha}_{ab}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle.\end{split}$$

Heuristically, this can be written distributionally as

$$\gamma \xi^b(x) \nabla^a T_{ab}(x)  \phi^{\alpha_1}(x_1) = - \delta(x-x_1) (\mathscr{L}_{\xi}\phi^{\alpha_1}(x_1)) + \nabla^a \left( \delta(x - x_1) \xi^b(x_1) M^{\alpha}_{ab}(x_1) \right),$$

though in this expression I haven't been very careful about volume factors in the delta functions; I claim there is some definition of the curved-space delta function for which this is the right expression!

2.3 Comments on dimensionful fields

I commented in the previous subsection that the Lie derivative of a tensor field $\phi^{\alpha}$ can always be written in terms of its covariant derivative as

$$(\mathscr{L}_{\xi} \phi)^{\alpha} = \xi^{a} \nabla_{a} \phi^{\alpha} + M^{\alpha}_{ab} \nabla^{a} \xi^b \tag{9}$$

for some tensor field $M^{\alpha}_{ab}.$ If $M^{\alpha}{}_{ab}$ is nonzero, then $\phi^{\alpha}$ is said to have some dimension, which so far is really just the statement that $\phi^{\alpha}$ is not a scalar field. Often the details of $M^{\alpha}_{ab}$ are broken down further to make more specific comments about the dimension — for example, if $M^{\alpha}_{ab}$ has an antisymmetric part, $M^{\alpha}_{[ab]} \neq 0$, then the field is said to have nonzero spin, because in this case $M^{\alpha}_{ab} \nabla^{a} \xi^b$ is nonzero for $\xi$ a field generating flat space rotations. If $M^{\alpha}_{ab} g^{ab}$ is nonzero then the field is said to have a nonzero scaling dimension.

There's an additional subtlety we have to keep in mind when discussing field dimensions, however, which is that it is sometimes convenient to think of a field as having a dimension that is different from that associated with the tensor $M^{\alpha}_{ab}$. What this means in practice is fixing a tensor $N^{\alpha}_{ab}$ that is locally constructed from the fields, and declaring that you want to study how the theory responds to the family of perturbations given by

$$\delta_{\xi} \phi^{\alpha} = - \xi^{a} \nabla_{a} \phi^{\alpha} - N^{\alpha}_{ab} \nabla^{a} \xi^b. \tag{10}$$

I think it's wrong to call the dimension specified by $N^{\alpha}_{ab} \neq M^{\alpha}_{ab}$ an intrinsic property of the field, which is the philosophical point of view often taken in textbooks. I find it more instructive to say that we can pick any tensor $N^{\alpha}_{ab}$ that we like and call it a dimension assignment or a choice of dimension for $\phi^{\alpha},$ and study how the theory responds to the associated family of perturbations. This is sometimes useful because, as in the example of the $D>2$ scalar field given in section 1, there are field theories where choosing to work with equation (10) instead of equation (9) gives an enhanced set of vector fields $\xi^a$ whose associated variations of symmetries of the theory; for example, a judicious choice of $N^{\alpha}_{ab}$ makes the field variation associated to scale transformations a symmetry of the free scalar in any dimension.

The one problem with choosing to study a dimension other than the standard Lie-derivative dimension is that it undermines the work we did in the previous two subsections to define the functional stress energy tensor. The functional stress energy tensor is useful in quantum field theories because, via equation (9), its correlation functions tell us how the theory responds to Lie derivatives of arbitrary compactly supported diffeomorphisms. But if we've decided that the field transformations we're interested in aren't Lie derivatives but some more general set given by equation (10), then this equation isn't so useful.

For this reason, when you choose to study non-Lie-derivative families of field transformations, it is very common to also change the definition of the stress energy tensor to match the family of transformations that interests you. The stress energy tensor is now defined not by equation (6), which expresses how the path integral weight changes under a variation of the metric, but by a generalized equation which expresses how the path integral weight changes under a particular coupled variation of the metric and the fields. In particular, for any metric variation $\delta g_{ab},$ we declare that we are interested in how the path integral responds to the total variation

$$g_{ab} \mapsto g_{ab} - \lambda \delta g_{ab},$$

$$\phi^{\alpha} \mapsto \phi^{\alpha} - \frac{\lambda}{2} (N-M)^{\alpha}_{ab} \delta g^{ab}.$$

This will induce a corresponding change in the path integral weight, and we declare the stress tensor to be given by the formula

$$\delta_{g \mapsto g - \lambda \delta g, \phi \mapsto \phi - \lambda (N-M) \delta g/2} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = - \frac{\gamma}{2} \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int_{\mathcal{M}} T_{ab} \delta g^{ab} \boldsymbol{\epsilon} \tag{11}.$$

 

Now, recall that for a compactly supported vector field $\xi^a,$ the path integral weight is invariant under the transformation $g \mapsto g - \lambda \mathscr{L}_{\xi} g,$ $\phi \mapsto \phi - \lambda \mathscr{L}_{\xi} \phi.$ We can think of this as two independent transformations that are applied simultaneously, one being given by

$$g_{ab} \mapsto g_{ab} - \lambda (\mathscr{L}_{\xi} g)_{ab},$$

$$\phi^{\alpha} \mapsto \phi^{\alpha} + \lambda (N-M)^{\alpha}_{ab} \nabla^{a} \xi^b,$$

and the other being given by

$$\phi^{\alpha} \mapsto \phi^{\alpha} - \lambda (\mathscr{L}_{\xi} \phi)^{\alpha} - \lambda (N - M)^{\alpha}_{ab} \nabla^{a} \xi^b = \phi^{\alpha} - \lambda \xi^{b} \nabla_{b} \phi^{\alpha} - \lambda N^{\alpha}_{ab} \nabla^a \xi^b.$$

The second transformation is just the family $\delta_{\xi}$ associated with our chosen field dimension $N^{\alpha}_{ab}.$ Since the combined transformation leaves the path integral weight unchanged, the response of the path integral weight to the second transformation must be the negative of the response of the path integral weight to the first transformation. I.e., we have

$$\delta_{\xi} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = - \delta_{g_{ab} \mapsto g_{ab} - \lambda (\mathscr{L}_{\xi} g)_{ab}, \phi^{\alpha} \mapsto \phi^{\alpha} - \frac{1}{2} \lambda (N - M)^{\alpha}_{ab} (\mathscr{L}_{\xi} g)^{ab}} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right).$$

In terms of the stress tensor defined by equation (11), this gives us

$$\delta_{\xi} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int_{\mathcal{M}} T_{ab} \nabla^{a} \xi^{b}. \tag{12}$$

This is just like equation (7), but for a different family of field variations, and a different definition of the stress tensor. All the Ward identities hold with the new stress tensor and the new family of field variations.

 

The takeaway of all this is that functional derivatives with respect to the metric give us a way of defining stress tensors, but it isn't always most useful to study the stress tensor defined by the independent functional derivative with respect to the metric. Depending on the problem we wish to solve, we might instead define the stress tensor by equation (11), and study the properties of the resulting operator.

 

Additionally, while I explained field dimensions in this subsection only in the context of quantum field theories, it should be clear how all these results work in classical field theories — we just remove the path integral weight from all expressions, and end up producing an equation like (5) instead of one like (7).

3. The current stress tensor

Since I was fairly careful in the last section about distinguishing between classical and quantum stress tensors, and since the considerations for understanding the differences between them are the same for $T_{ab}$ as for $T^{C}_{ab},$ I will focus in this section on quantum field theories.

Suppose we have a quantum field theory defined in flat space, with flat coordinates $x^1, \dots, x^D.$ It is assumed that the response of the path integral to field variations is local, meaning that for a field variation $\phi^{\alpha} \mapsto \phi^{\alpha} - \lambda \delta \phi^{\alpha}$ we have

  $$\delta \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int d^{D} x P_{\alpha} \delta \phi^{\alpha}$$

for some tensor $P_{\alpha}$ locally constructed from the fields. As always, this should only hold for field variations that are well-enough behaved at infinity that potential surface terms can be neglected.

 

If the fields $\phi^{\alpha}$ are varied by the perturbation

$$\phi^{\alpha} \mapsto \phi^{\alpha} - \lambda \partial_{\nu} \phi^{\alpha}$$

for some fixed coordinate $\nu$, then the path integral measure changes as

$$\delta \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = - \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int d^{D} x P_{\alpha} \partial_{\nu} \phi^{\alpha}.$$

Translation in the $\nu$ direction is said to be a local symmetry of the theory if the corresponding change in the path integral weight is the integral of a total derivative. I.e., it is a local symmetry of the theory if we have

$$P_{\alpha} \partial_{\nu} \phi^{\alpha} = \partial^{\mu} T^C_{\mu \nu} \tag{13}$$

for some vector $T^C_{\mu \nu}.$ (Remember, for the moment this is a vector, because $\nu$ is fixed, so only $\mu$ ranges over the coordinates.) By repeating this process for all values of $\nu$, we get a full list of numbers $T^{C}_{\mu \nu}$ that is called a current stress tensor of the theory. For the moment there are some ambiguities in the definition of $T^{C}_{\mu \nu}$; it can be modified by a term whose divergence is zero without affecting any of the equations we've written so far.

We now ask, as is traditional in derivations of Noether's theorem: what if we smear out our symmetry? That is, what if we pick a smooth function $\rho(x)$ and consider the field variation

$$\phi^{\alpha}(x) \mapsto \phi^{\alpha}(x) - \lambda \rho(x) \partial_{\nu} \phi^{\alpha}(x)?$$

So long as $\rho(x)$ is sufficiently well behaved at infinity — in particular, if it is compactly supported — we have

$$\delta_{\rho} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = - \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int d^{D} x \rho P_{\alpha} \partial_{\nu} \phi^{\alpha}.$$

Plugging in equation (13), we obtain the formula

$$\delta_{\rho} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = - \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int d^{D} x \rho \partial^{\mu} T^{C}_{\mu \nu}. \tag{14}$$

Since any vector field on flat space can be written as a linear combination of the coordinate vector fields, equation (14) gives us a formula for how the theory responds to an arbitrary (well-behaved-at-infinity) local diffeomorphism. This formula is

$$\delta_{\xi} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = - \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \int d^D x\, \xi^{\nu} \partial^{\mu} T^C_{\mu \nu}.$$

This looks just like equation (12) from the previous section, or a partially-integrated version thereof, but with one important difference: the field transformations $\delta_{\xi}$ are different.

 

To be precise, we will denote by $\delta_{\xi, F}$ the field transformation

$$\phi^{\alpha} \mapsto \phi^{\alpha} - \lambda \xi^{\mu} \partial_{\mu} \phi^{\alpha} - \lambda N^{\alpha}_{\mu \nu} \partial^{\mu} \xi{\nu},$$

where $N^{\alpha}_{\mu \nu}$ is some tensor controlling the dimensions of the fields. We will denote by $\delta_{\xi, C}$ the field transformation

$$\phi^{\alpha} \mapsto \phi^{\alpha} - \lambda \xi^{\mu} \partial_{\mu} \phi^{\alpha}.$$

The fundamental difference between the current stress tensor and the functional stress tensor is that they satisfy

$$\delta_{\xi, F} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \xi^{\nu} \partial^{\mu} T_{\mu \nu}^F \tag{15}$$

and

$$\delta_{\xi, C} \left( \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \right) = \gamma \mathcal{D} \phi^{\alpha} e^{\gamma S[\phi]} \xi^{\nu} \partial^{\mu} T_{\mu \nu}^C. \tag{16}$$

In these expressions and henceforth, I have added the superscript "F" to the functional stress tensor to make it clearly distinct from $T_{\mu \nu}^{C}.$ Because the transformations $\delta_{\xi, F}$ and $\delta_{\xi, C}$ are different for dimensionful fields, the functional and current stress tensors of these fields are different.

Often, people say that the current stress tensor $T_{\mu \nu}^C$ is ambiguous up to the addition of "improvement terms." In particular, one can add to $T_{\mu \nu}^C$ a term whose $\mu$-divergence vanishes identically on all field configurations. As I mentioned in section 1 of my post on Ward identities, it was proven by Wald that if a tensor locally constructed from the fields has vanishing divergence on all field configurations, then it is itself a total divergence. I.e., the only terms we can add to $T_{\mu \nu}^C$ without affecting equation (16) are of the form

$$\partial^{\rho} j_{\rho \mu \nu},$$

where $j_{\rho \mu \nu}$ is locally constructed from the fields and antisymmetric in the first two indices. Terms like this are often added to the classical current stress tensor, for example, to make it symmetric or, in the case of conformally invariant field theories, traceless.

However, because the addition of such terms does not affect equation (16), there is no "total-divergence" improvement term that can make $T^{C}_{\mu \nu}$ equal to $T^{F}_{\mu \nu}$ at the quantum level for dimensionful fields. While it may be possible to do this at the classical level, i.e., to make $T^{C}_{\mu \nu}$ and $T^{F}_{\mu \nu}$ agree on field configurations that satisfy the equations of motion, when equations (15) and (16) are different, the Ward identities of $T^{C}_{\mu \nu}$ and $T^{F}_{\mu \nu}$ will always be different no matter what total-divergence improvement term is added to $T^{C}_{\mu \nu}.$

As an example, think for a moment about two-dimensional Euclidean quantum field theory in flat space. Suppose that $\phi^{\alpha}$ is a field that transforms under the rotation

$$\xi^{\mu} = - y \left( \frac{\partial}{\partial x} \right)^{\mu} + x \left( \frac{\partial}{\partial y} \right)^{\mu}$$

as

$$\delta \phi^{\alpha} = - \xi^{\mu} \partial_{\mu} \phi^{\alpha} - s^{\alpha}{}_{\beta} \phi^{\alpha}$$

for some antisymmetric matrix $s.$ Suppose further that this transformation is a symmetry of the theory. The Ward identities for $n$-point functions of $\phi$ for the two different stress tensors can be written as

$$\begin{split} & \langle \int d^2 x T^{F}_{\mu \nu} \partial^{\mu} \xi^{\nu} \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle \\ & = - \sum_{j=1}^{n} \left( \xi^{\mu} \partial_{\mu, j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle + s^{\alpha_j}{}_{\beta_j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\beta_j}(x_j) \dots \phi^{\alpha_n}(x_n) \rangle \right) \end{split} \tag{17}$$

and

$$\langle \int d^2 x T^{C}_{\mu \nu} \partial^{\mu} \xi^{\nu} \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle = - \sum_{j=1}^{n} \xi^{\mu} \partial_{\mu, j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle. \tag{18}$$

Because the total field transformation is a symmetry of the theory, the right-hand side of (17) must vanish. This gives us the identities

$$\langle \int d^2 x T^{F}_{\mu \nu} \partial^{\mu} \xi^{\nu} \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle = 0$$
and
$$\langle \int d^2 x T^{C}_{\mu \nu} \partial^{\mu} \xi^{\nu} \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle = \sum_{j=1}^{n} s^{\alpha_j}{}_{\beta_j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\beta_j}(x_j) \dots \phi^{\alpha_n}(x_n) \rangle.$$

Since $\xi^{\mu}$ is a rotation, we have $\partial^{\mu} \xi^{\nu} = \epsilon^{\mu \nu}.$ Plugging this in, we obtain

$$\langle \int d^2 x (T^{F}_{1 2} - T^{F}_{2 1}) \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle = 0$$

and

$$\langle \int d^2 x (T^{C}_{1 2} - T^{C}_{2 1}) \phi^{\alpha_1}(x_1) \dots \phi^{\alpha_n}(x_n) \rangle = \sum_{j=1}^{n} s^{\alpha_j}{}_{\beta_j} \langle \phi^{\alpha_1}(x_1) \dots \phi^{\beta_j}(x_j) \dots \phi^{\alpha_n}(x_n) \rangle.$$

So $T^{F}_{12} - T^{F}_{2 1}$ has no contact terms in correlation functions, meaning the tensor $T^{F}_{\mu \nu}$ is identically symmetric, i.e., symmetric on all field configurations. By contrast, the operator $T^{C}_{1 2} - T^{C}_{2 1}$ does have contact terms in correlation functions, meaning that even if some total-divergence improvement term $\partial^{\rho} j_{\rho \mu \nu}$ is chosen to make $T^{C}_{\mu \nu}$ symmetric classically, i.e., on field configurations satisfying the equations of motion, it will not be identically zero as a quantum object. (Unless, by some miracle, the transformation $\delta \phi^{\alpha} = - s^{\alpha}{}_{\beta} \phi^{\alpha}$ is itself a symmetry of the theory, in addition to the full symmetry $\delta \phi^{\alpha} = - \xi^{\mu} \partial_{\mu} \phi^{\alpha} - s^{\alpha}{}_{\beta} \phi^{\alpha}.$)

Incidentally, the same issue arises in conformal field theory, where $T^{F}_{\mu \nu}$ is identically traceless while the trace of $T^{C}_{\mu \nu}$ has contact terms given by the scaling dimensions of operators.

Supposedly, there is a way to add terms to $T^{C}_{\mu \nu}$ to make it equal to $T^{F}_{\mu \nu}$ in a general quantum field theory, at least one with enough symmetry, but these terms cannot be total-divergence improvement terms. They must instead be tensors locally constructed from the fields whose divergence vanishes on classical configurations so as to preserve the operator equation $\partial^{\mu} T^{C}_{\mu \nu} = 0,$ but not on all field configurations so as to allow the Ward identity (16) to be modified into the Ward identity (15). Some systematics on constructing these "non-divergence" improvement terms are discussed in the paper by Kourkoulou, Nicolis, and Sun that I mentioned in the introduction.