Zeta function regularization, heat kernels, and one-loop determinants

In the path integral formulation of quantum field theory, a free theory is one whose Euclidean action can be expressed as
$$S[\phi] = \frac{1}{2} \int_{\mathcal{M}} \boldsymbol{\epsilon} \phi L \phi, \tag{1}$$
where $\phi$ is some tensor field and $L$ is a self-adjoint differential operator whose spectrum is bounded below. For example, if we make $\phi$ a real scalar field and $L$ the differential operator
$$L = m^2 - \nabla^2,$$
then $S[\phi]$ is the action of a free scalar with mass $m.$ It is useful to have general techniques for studying the path integrals of free field theories, not only because free theories serve as an important testing ground for interesting physical ideas, but also because in the semiclassical limit, any quantum field theory can be approximated as free. In particular, if we consider some Euclidean quantum field theory defined by the path integral
$$\int \mathcal{D} \phi e^{- \frac{1}{\hbar} S[\phi]},$$
then in the limit $\hbar \rightarrow 0$ the path integral can be approximated as a sum over stationary points of the action — these are the classical configurations of the theory — together with a perturbative-in-$\hbar$ series of quantum corrections, the path integral of fluctuations about the classical configurations. The first nontrivial term in this expansion will be quadratic in the field perturbations, since the linear term must vanish due to the assumption that the action is stationary. By applying integration by parts with appropriate boundary conditions, a quadratic functional of the fields can always be put in the form (1), with $\phi$ replaced by a linearized perturbation $\delta \phi$.

This note explains the notion of a one-loop determinant, which is a way of thinking of the path integral of a free theory as computing the (renormalized) determinant of a differential operator. It discusses how to renormalize this determinant using zeta function regularization, and how to compute that regularized determinant in the abstract using a heat kernel.

In section 1, I show how the path integral of a free theory can be interpreted as computing the determinant of a differential operator, which explains the terminology "one-loop determinant." (The term "one-loop" comes from the fact that in the semiclassical limit, the "free theory approximation" shows up at the first nontrivial order in $\hbar$.)

In section 2, I introduce zeta function regularization and show how it is related to the heat kernel $K(t)$ of $L.$

In section 3, I use zeta function regularization to compute the partition function of the quantum mechanical harmonic oscillator.

In section 4, I give some mathematical comments about the asymptotics of the heat kernel, and explain what I think are the failings of the (sort-of-incorrect) formula that is usually written down,
$$Z = \exp\left[ \frac{1}{2} \int_{0}^{\infty} dt\, \frac{1}{t} K(t) \right].$$
I also comment briefly on the "physicalness" of zeta function regularization by relating it to other regularization techniques, and refer to a blog post of Terry Tao for further information.

I learned this material from slightly unconventional sources. I learned about zeta function regularization and heat kernels from Stephen Hawking's essay in this "Einstein centenary" collection. I think I learned how to apply zeta function regularization to the harmonic oscillator from this paper, but I'm not sure; I learned this material >2 years ago and am adapting this post from notes I took at the time. While fixing some details in this post I read Hawking's original paper on the subject, which is fantastic. I found section 2 of this paper helpful for understanding how to think about Hawking's infamous number $\mu$ that appears in the path integral measure. I also highly recommend Mark Kac's delightful paper "Can one hear the shape of a drum?", which gives a great, intuitive explanation of the asymptotic properties of the heat kernel.

Prerequisites: Path integral formulation of quantum field theory. The discussion of asymptotic properties of the heat kernel in section 4 uses abstract index notation, and requires knowing enough differential geometry to understand covariant derivatives. You don't need to know differential forms to understand this post, but I'll always write $\int_{\mathcal{M}} \boldsymbol{\epsilon}$ instead of $\int d^d x \sqrt{g}.$

1. Path integrals and determinants

Suppose we have a Euclidean field theory defined by the action $S[\phi]$ on the manifold $\mathcal{M}$ with metric $g_{ab}.$ For simplicity, we'll assume the fields are real, but there's no major conceptual difference in working with theories of complex fields. The action is assumed to be free, meaning it can be written in the form
$$S[\phi] = \frac{1}{2} \int_{\mathcal{M}} \boldsymbol{\epsilon} \phi L \phi$$
for some self-adjoint differential operator $L$ whose spectrum is bounded below. We want to evaluate the partition function of the theory, which is the path integral
$$Z = \int \mathcal{D}\phi e^{- S[\phi]}.$$
The trick will be to use the fact that $L$ is self-adjoint to find an orthonormal eigenbasis of field configurations, i.e., a set of field configurations $\{\phi_n\}$ satisfying
$$L \phi_n = \lambda_n \phi_n$$
and
$$\int_{\mathcal{M}} \boldsymbol{\epsilon} \phi_n \phi_m = \delta_{n m}.$$
If we expand a generic field configuration as
$$\phi = \sum_n c_n \phi_n,$$
then it seems natural to think of the path integral measure $\mathcal{D}\phi$ as being expressible as
$$\mathcal{D}\phi = \prod_{n} \int d c_n.$$
In fact, there are times when interpreting the path integral measure this way gives inconsistent physical answers. Many of these are ameliorated by instead writing
$$\mathcal{D}\phi = \prod_{n} \frac{1}{\sqrt{2 \pi \mu}} \int d c_n,$$
with $\mu$ an as-yet unspecified constant, and fixing $\mu$ at the end of the calculation by imposing an appropriate consistency condition. I don't think it should be so surprising that such a modification is occasionally necessary. There's no real reason to think the path integral measure, this unwieldy behemoth that we don't understand on rigorous footing, should be so simple as $\mathcal{D} \phi = \prod_n \int d c_n.$ We should allow ourselves some freedom in that identification; adding a constant $\mu$ to each mode integral is a simple way to do that which is "ultralocal" in frequency space, and it turns out to be enough in many cases to get useful and interesting physical answers.

If we expand the action in terms of the eigenbasis $\{\phi_n\},$ we find
$$S[\phi] = \frac{1}{2} \sum_{m, n} c_m c_n \lambda_n \int_{\mathcal{M}} \boldsymbol{\epsilon} \phi_m \phi_n = \frac{1}{2} \sum_{n} \lambda_n c_n^2.$$
The path integral
$$\int \mathcal{D}\phi e^{- S[\phi]}$$
then turns into a product of Gaussian integrals for all nonzero eigenvalues, plus some overall prefactor corresponding to the zero eigenvalues. We have
$$\int \mathcal{D}\phi e^{- S[\phi]} = \left(\frac{1}{\sqrt{2 \pi \mu}}\right)^{\text{# zero modes}} \operatorname{Vol}(\text{zero modes}) \prod_{n} \frac{1}{\sqrt{2 \pi \mu}} \int d c_n e^{- \frac{1}{2} \lambda_n c_n^2}. \tag{2}$$
Here $\operatorname{Vol}(\text{zero modes})$ means the volume of the domains of the coefficients of the "zero modes," i.e., the eigenfunctions $\phi_n$ with eigenvalue zero. There may be no such eigenfunctions, in which case this factor is set to $1.$ There may be zero modes whose coefficients take arbitrary values, in which case the corresponding prefactor is thought of as an unphysical divergence that should be renormalized away. In some cases, though, the prefactor $\operatorname{Vol}(\text{zero modes})$ is a finite quantity that depends on the geometry of spacetime in a nontrivial way; in these cases, the zero-mode prefactor makes an essential contribution to the path integral. We won't see an example of that in this post, but maybe in a future one; these zero modes play an essential role, for example, in establishing higher-genus modular invariance of the two-dimensional compact boson CFT.

By the way, generally speaking the prefactor $\sqrt{2 \pi \mu}^{- \text{# zero modes}}$ is physically unimportant. If the number $\mu$ doesn't depend on the geometry or on the parameters of the theory, and if the number of zero modes is a topological invariant, which is often the case, then $\sqrt{2 \pi \mu}^{-\text{# zero modes}}$ has no dependence on the parameters of the theory and therefore has no physical meaning; it's like a constant shift of the ground state energy of the theory.

The Gaussian integrals in (2) can be evaluated in the standard way, giving the result
$$\int \mathcal{D} \phi e^{- S[\phi]} = \left(\frac{1}{\sqrt{2 \pi \mu}}\right)^{\text{# zero modes}} \operatorname{Vol}(\text{zero modes}) \prod_{n} \frac{1}{\sqrt{\mu \lambda_n}}.$$
Here the product ranges over all nonzero eigenvalues of $L.$ Formally, this can be rewritten as
$$\left(\frac{1}{\sqrt{2 \pi \mu}}\right)^{\text{# zero modes}} \operatorname{Vol}(\text{zero modes}) \det \left(\mu L_{\times} \right)^{-1/2},$$
where the notation $L_{\times}$ refers to the restriction of $L$ to its nonzero eigenvectors. We will drop this notation going forward, though, and just ask the reader to keep in mind that whenever we take a determinant of $L$, or write any function of its eigenvalues, we really mean only the nonzero eigenvalues.

The differential operator $\mu L_{\times}$ is unbounded, so it doesn't actually have a determinant. But there's a convenient and universal way to assign a finite value to this determinant, which seems to have useful physical consequences. This is the subject of the next section.

One comment, though, before we proceed: everything I've written so far, and everything I'll write below, assumes that the eigenvalue spectrum of $L$ is discrete. This is a feature of compact manifolds. For non-compact manifolds, the eigenvalue spectrum is often continuous, but also the assumption made above that a generic field configuration can be written as $\phi(x) = \sum_{n} c_n \phi_n(x)$ for arbitrary values of $c_n$ is false; boundary conditions on the field, like requiring it to fall off at a certain rate near infinity, constrain the $c_n$ coefficients relative to one another. This is one of many "infrared" issues that appear in trying to study quantum field theory on non-compact manifolds. One hopes, however, that non-compact manifolds of interest can be achieved as limits of compact manifolds, and that the properties of a quantum theory on a non-compact manifold can be computed as limits of properties of that theory on compact manifolds; this is what happens, for example, when you study quantum field theory in Minkowski spacetime by defining it on a spacelike sphere or torus and taking the limit as the size of that sphere/torus goes to infinity.

2. Zeta function regularization

We will now start assuming that $L$ is a positive semidefinite operator, i.e., that all of its nonzero eigenvalues are positive. This is not an especially significant restriction, as any operator whose spectrum is bounded below can be made positive semidefinite by a constant shift.

Our goal is to assign a finite value to the formal expression
$$\det\left( \mu L \right) = \prod_{n} \mu \lambda_n.$$
If this product were convergent, then we could take its logarithm and obtain
$$\log\det\left(\mu L \right) = \sum_n \log\left(\mu \lambda_n \right). \tag{3}$$
A convenient way to rewrite the logarithm of a number is as the linear contribution to its power function, i.e., as
$$\log(x) = - \left. \frac{d}{ds} x^{-s} \right|_{s=0}.$$
Substituting this into equation (3) gives
$$\log\det\left(\mu L \right) = - \sum_n \frac{d}{ds} \left. (\mu \lambda_n)^{-s} \right|_{s=0}.$$
If this series were uniformly convergent, then we could exchange the sum and the derivative to obtain
$$\log\det\left(\mu L \right) = - \left. \frac{d}{ds} \left( \sum_n (\mu \lambda_n)^{-s} \right)\right|_{s=0}.$$
Consider now $s$ as a complex number. For values of $s$ with sufficiently large real part, unless the eigenvalues $\{\lambda_n\}$ are particularly badly behaved at large values of $n,$ the sum
$$\sum_n (\mu \lambda_n)^{-s}$$
converges. In fact, it converges uniformly on compact subsets in the $s$-plane, and therefore by Morera's theorem is an analytic function on its domain of definition. On this domain, we define the $\zeta$-function
$$\zeta_{\mu L}(s) = \sum_{n} (\mu \lambda_n)^{-s}.$$
It can be extended to other parts of the complex plane by analytic continuation. If this extension is unique and regular at $s=0$, then we can formally define the determinant of $\mu L$ by
$$\det\left(\mu L \right) = e^{- \zeta_{\mu L}'(0)}.$$

Supposedly, one can show that when $L$ is a so-called elliptic differential operator, the asymptotic growth of the eigenvalues for large $n$ is such that the $\zeta$ series is guaranteed to converge uniformly for $\operatorname{Re}(s)$ above some threshold, and such that the analytic continuation is guaranteed to be regular at $s=0.$ For this result, Hawking cites the essay "Complex powers of an elliptic operator" in this book, but the paper is pretty technical and I haven't been able to understand it. No big deal, though; we can check that the eigenvalues of any particular operator we're interested in admit an appropriate analytic continuation, and indeed we'll do this for the harmonic oscillator in section 3. In fact, we'll see in section 4 that the question of whether the $\zeta$ series converges for large enough $\operatorname{Re}(s)$ can be framed in terms of the asymptotic behavior of a particular function of the eigenvalues, and give an intuitive explanation for why we should expect it to converge.

In the zeta function's domain of definition, we can write
$$\zeta_{\mu L}(s) = \mu^{-s} \sum_{n} \lambda_n^{-s} = \mu^{-s} \zeta_L(s).$$
Because $\mu^{-s}$ is regular everywhere in the $s$ plane, this expression holds everywhere. So in particular, near $s=0,$ we have
$$\zeta_{\mu L}(s) = (1 - s \log\mu) \zeta_{L}(s),$$
and hence
$$\zeta_{\mu L}'(0) = - \zeta_{L}(0) \log\mu + \zeta_{L}'(0).$$
It is often the case, though not always, that $\zeta_{L}(0)$ vanishes; when this happens, we have $\zeta_{\mu L}'(0) = \zeta_{L}'(0),$ and the normalization constant $\mu$ makes no physical contribution to the path integral. In any case, it is clear that we only need to study the function $\zeta_L(s)$ in a neighborhood of $s=0,$ since $\zeta_{\mu L}'(0)$ is completely determined by $\zeta_{L}(0),$ $\zeta_{L}'(0),$ and $\mu.$

There is a useful way to rewrite the function $\zeta_{L}(s)$, which is to insert the gamma function
$$\Gamma(s) = \int_{0}^{\infty} dt\, t^{s-1} e^{-t}.$$
We can write
$$\zeta_{L}(s) = \frac{1}{\Gamma(s)} \sum_{n} \int_{0}^{\infty} dt\, t^{s-1} e^{-t} \lambda_n^{-s}.$$
Making the substitution $t \mapsto t \lambda_n$ gives
$$\zeta_{L}(s) = \frac{1}{\Gamma(s)} \sum_{n} \int_{0}^{\infty} dt\, t^{s-1} e^{-t \lambda_n}.$$
For well-behaved $\lambda_n$ and sufficiently large values of $\operatorname{Re}(s),$ the sum $\sum_n t^{s-1} e^{-t \lambda_n}$ converges uniformly on compact subsets of $t$, which means we can interchange the sum and the integral to write
$$\zeta_L(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} dt\, t^{s-1} \sum_n e^{-t \lambda_n}.$$
(The only real restriction to guarantee this uniform convergence is that $\operatorname{Re}(s)$ needs to be large enough to cancel any small-$t$ poles in $\sum_n e^{-t \lambda_n}$; more on that in section 4.)

The sum $\sum_n e^{-t \lambda_n}$ can in principle be evaluated directly if we know all of the eigenvalues of the operator $L,$ which will then let us evaluate $\zeta_L(s)$ near $s=0.$ Indeed, we'll do this explicitly in section 3 for the harmonic oscillator partition function. But it isn't always the case that we know all of the eigenvalues, or how to sum them, or how to understand the integral of $t^{s-1}$ against that sum. But we don't actually need all of the eigenvalues to evaluate $\zeta_L(s)$; we only need the sum. The total sum has an elegant relationship to the heat kernel of the operator $L,$ which in some cases can be studied using techniques from partial differential equations even if we can't study the eigenvalues directly.

The idea is to write the sum $\sum_n e^{- t \lambda_n}$ in terms of an orthonormal basis of eigenfunctions $\{\phi_n\}$ as
$$\sum_n e^{-t \lambda_n} = \int_{\mathcal{M}, x} \boldsymbol{\epsilon} \sum_n e^{- t \lambda_n} \phi_n(x) \phi_n(x).$$
If we define the function $K(x, x', t)$ by
$$K(x, x', t) = \sum_n e^{- t \lambda_n} \phi_n(x) \phi_n(x'),$$
then we can express this as
$$\sum_n e^{- t \lambda_n} \int_{\mathcal{M}, x} \boldsymbol{\epsilon} K(x, x, t).$$
The function $K(x, x', t)$ for $t > 0$ is interesting because it solves the heat equation
$$\frac{\partial K(x, x', t)}{\partial t} + L_{x} K(x, x', t) = 0,$$
which is straightforward to check. The heat equation is linear and involves only a single derivative in $t,$ so a solution of the heat equation is determined uniquely for all $t > 0$ by its boundary condition at $t=0.$ In the limit $t \rightarrow 0,$ the function $K(x, x', t)$ limits to
$$K(x, x', 0) = \sum_n \phi_n(x) \phi_n(x') = \delta(x, x'),$$
where the delta function is normalized with respect to the metric, so that it satisfies
$$\int_{\mathcal{M}, x} \boldsymbol{\epsilon} \delta(x, x') f(x) = f(x').$$

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

Wigner's theorem

Wigner's theorem is one of the most fundamental results in quantum theory, but I somehow didn't hear of it for the first time until the third year of my PhD. Even then, it took me another year or so to fully appreciate the theorem's importance. I suspect this experience is common — Wigner's theorem is thought of as being fairly technical or mathematical, and doesn't get covered in most quantum mechanics courses. But because it's so essential, I'd like to dedicate a post to explaining and proving it. The statement of the theorem is simple: every symmetry of a quantum system can be represented as a unitary or anti-unitary operator on Hilbert space . (Here we are implicitly thinking about quantum states as vectors in a Hilbert space — there are ways of thinking about Wigner's theorem from a more operator-algebraic point of view, but the Hilbert space picture is a good place to start.) The reason Wigner's theorem is so valuable is that if we believe a sy