Structural Aspects of the Exponential Expansion of the Heat Kernel

G. Kennedy Department of Mathematics Glasgow Caledonian University Glasgow G4 0BA, United Kingdom and S.A. Fulling Department of Mathematics Texas A&M University College Station, Texas 77843-3368, USA

Abstract. (1) The small-time expansion of the logarithm of the heat kernel of a scalar operator contains only terms that are “connected” by tensor contractions. We offer a simple proof of this fact based on the necessary structure of a heat kernel on a product manifold. This result generalizes and systematizes previous work, including the connectedness property of the graph expansions of Osborn and coworkers and the observation of Parker and coworkers about the role of the curvature scalar.

(2) For a matrix-valued (vector bundle) operator the expansion does not exhibit such an obvious connectedness property. However, we show, using a theorem of Friedrichs in the representation theory of Lie algebras, that an analogous property does persist in the bundle case when tensor contractions are augmented by Lie products. (3) It follows that the terms of the small-time expansion of the heat kernel itself may be decomposed into a direct sum of connected and disconnected components, and that explicit information on the highly disconnected components at any order is provided by only the low-order terms in the expansion of the logarithm.

Heat Kernel Techniques and Quantum Gravity, ed. by S. A. Fulling, Discourses in Mathematics and Its Applications, No. 4, \begin{matrix}c \\ ◯ \end{matrix} Department of Mathematics, Texas A&M University, College Station, Texas, 1995, pp. 101–114.

Physical applications, particularly those concerning the analysis of renormalizability and other high-energy properties of quantum field theories, have stimulated extensive explicit calculations of the small-time asymptotic behavior of the heat kernel of a generic second-order linear differential operator. In the case of an operator on scalar functions, it has been observed [6, 10, 22, 24, 21, 3, 23] that noteworthy simplifications are achieved by “pushing the series up into the exponent” (see (2) below). Other authors have given special attention to the role of the curvature scalar, R, in this position [8, 9, 26, 15]. Attempts to extend this approach to operators on vector fields (in the generic sense) have not been very successful, however, and it seems to be a common opinion that no such exponential expansion exists in that case (even formally), or at least that there is no useful algorithm for calculating the terms.

The most striking feature of the scalar exponential expansion is that all its terms are “connected” by tensor contractions. That is, terms such as R, R^{αβγδ}R_{αβγδ}, and V_{;μ}V_{;}^{μ} appear, but terms such as R^{2} and V V_{;μ}^{μ} are absent (where R^{····} is the curvature tensor, V is a scalar potential, and the semicolon denotes covariant differentiation). It follows that the exponential expansion contains the information of the conventional expansion (1) in a more elegant and compact form.

The present communication makes three main points. For one thing, we present a “soft” proof of this connected structure, based on the functorial behavior of the heat kernel under products of manifolds, rather than on a detailed examination of recursion relations or Dyson series. The theorem we prove is general and is not restricted to product systems; products are used merely as a tool in the proof. Second, we establish the generalization of the connectedness property to the case of “minimal” or Laplacian-like operators on vector bundles. There the exponential series contains not only terms that are connected in the tensorial sense, but also Lie products of the basic elements. A theorem due to Friedrichs in the representation theory of Lie algebras enables one to make this into a precise, positive statement about which objects are sufficiently “monolithic” to be allowed in the exponential series. Third, we observe that this connectedness property of the exponential expansion implies that the terms of the conventional expansion (1) may be decomposed into a direct sum of connected and disconnected components. Explicit information on the highly disconnected components of the conventional expansion at any order is therefore available, in principle, from a knowledge of only the low-order terms in the exponential expansion.

Our presentation is brief; full proofs will be published elsewhere. The theorem describes the overall algebraic structure of the exponential series, but does not prescribe an efficient algorithm for computing it, much less an explicit formula for the general term comparable to the connected-graph formulas found for special (scalar) cases by Osborn et al. [10, 22, 21, 3, 23]. There is plenty of scope for future work in that direction.

We will be working on a C^{∞} paracompact real manifold M of finite dimension d which serves as the base manifold of a C^{∞} real or complex finite-dimensional vector bundle E. We will assume that both T^{∗}(M) and E are equipped with C^{∞} connections ∇ and that these extend to dual and product bundles in the usual way. The reader is referred to [18] or [14] for background differential geometry and to [20] for details of induced bundles, induced connections, and the like, which will be introduced shortly.

A second-order differential operator A: Γ^{∞}(E) → Γ^{∞}(E) which maps the space of smooth sections of E into itself will be represented as A = ∑_{r=0}^{2} A_{r}∇^{2−r}, where, for each r ∈ {0, 1, 2}, the coefficient tensor A_{r} ∈ Γ^{∞} (End(E) ⊗ S(⊗^{2−r}T (M))), S denoting symmetrization, and contraction with ∇^{2−r} is understood. We will further assume that A has scalar principal symbol, i.e., that A_{0} is a tensor product of the identity endomorphism, I, of the bundle with an element of Γ^{∞} (S(T (M) ⊗ T (M))), also denoted by A_{0} , and that A is a strongly elliptic operator, in which case A_{0} is invertible with inverse A^{0} ∈ Γ^{∞} (S(T^{∗}(M) ⊗ T^{∗}(M))), the negative of which plays the role of a natural metric on M.

In such circumstances, as is well known, the restriction to the diagonal of the integral kernel of the heat operator e^{−tA} associated to A admits an asymptotic expansion of the form [13, 12]

K(x, t) ∼ (4πt)^{− \frac{d}{2}} ∑_{r=0}^{∞} a_{r}(x) t^{r}  (t → 0^{+}),  (1)

where x ∈ M, a_{0} = I, and each a_{r} ∈ Γ^{∞}(End(E)). Explicit expressions for a_{1} , a_{2} , a_{3} , and a_{4} have been obtained by a number of authors [29, 5, 19, 27, 11, 4, 12, 25, 1, 2]. In general, a_{r} is a linear combination of monomial invariants each of which is formed by tensor contraction of a monomial element of the associative algebra generated by the fundamental tensors present in the problem: the torsion tensor, T, the curvature tensor, R, the bundle curvature, R, the coefficient tensors, A_{0} , A_{1} , and A_{2} , of the differential operator, covariant derivatives of each of these tensors, and the inverse, A^{0}, of the leading-order coefficient. There is, a priori, no restriction on the allowed tensor contractions, so a wide variety of monomial invariants is available at each order r.

Here, however, we are more interested in the alternative small-time asymptotic expansion

K(x, t) ∼ (4πt)^{− \frac{d}{2}} exp [∑_{r=1}^{∞} c_{r}(x) t^{r}]  (t → 0^{+}),  (2)

or, more precisely,

log [(4πt)^{\frac{d}{2}} K(x, t)] ∼ ∑_{r=1}^{∞} c_{r}(x) t^{r}  (t → 0^{+}),  (3)

where each c_{r} ∈ Γ^{∞}(End(E)). Although perhaps less well known, the existence of the asymptotic expansion (3) for the logarithm of the heat kernel follows readily from that of (1). We refer to (2) or (3) as the exponential expansion of the heat kernel and to (1) as the conventional expansion.

At first sight it may appear that the exponential expansion offers nothing new, but instead merely presents the information that is already contained in the conventional expansion in a repackaged form: one that is more compact but possibly less convenient for calculations. Recall, however, that in practice we have explicit knowledge only of the first few terms in either expansion, and these contain qualitatively different information. Perhaps the easiest way to see this is to imagine expanding the exponential in (2) as a Maclaurin series and comparing the result with (1): clearly, c_{1}, for example, contributes to each a_{r} for all orders r ≥ 1.

There is a more than superficial analogy here with the 1/N expansion of, for example, an O(N)-invariant quantum field theory [28]. The usual loop expansion of the effective action is an expansion in powers of h̄; in the 1/N expansion the Feynman diagrams contributing to the loop expansion are summed to all orders in h̄ according to their large-N behavior. The 1/N expansion provides an alternative to expansion in powers of h̄ that is more appropriate for addressing problems of a nonperturbative nature, such as phase transitions and critical phenomena [7, 17]. Although we will not pursue the point here, in the analogy with the exponential expansion the manifold dimension d essentially plays the role of N, so the exponential expansion can be regarded as a kind of large-dimension expansion of the heat kernel.

The so-called summed form of the heat kernel expansion due to Parker and coworkers [26, 15] lies somewhere between the conventional expansion (1) and the exponential expansion (2). Indeed, it was an attempt to understand the properties of that expansion which led to the investigation reported here. Properties of the summed expansion easily follow from those of the exponential expansion.

Of course, the conventional and exponential expansions are closely related. It is an easy exercise to verify from (1)–(3) that the respective coefficients satisfy the relationships

a_{r} = ∑_{q=1}^{r} \frac{1}{q!} ∑_{s_{1}≥1,...,s_{q}≥1}^{s_{1}+···+s_{q}=r} c_{s_{1}} ⋯ c_{s_{q}}  (4)

for r > 0 and

c_{r} = ∑_{q=1}^{r} \frac{(−1)^{q+1}}{q} ∑_{s_{1}≥1,...,s_{q}≥1}^{s_{1}+···+s_{q}=r} a_{s_{1}} ⋯ a_{s_{q}} .  (5)

In particular, it follows immediately from (5) that, like the corresponding a_{r}, each c_{r} is a linear combination of monomial invariants each of which is formed by tensor contraction of a monomial element of the associative algebra generated by the fundamental tensors listed earlier. This is not surprising. What is perhaps surprising are the following facts:

• Each c_{r} can be written as a linear combination of monomial invariants each of which is formed from a product of elements of the Lie algebra generated by the fundamental tensors listed earlier by means of a particular type of tensor contraction which permits interpretation as a connected graph.

• In view of the connected-graph interpretation of the c_{r}, the expansion of a_{r} in (4) may be rewritten as a direct sum of connected and disconnected components.

These are our main results and will be made more precise in due course. For the present, note that the first demonstrates that the c_{r} possess a simplicity in algebraic structure which is not evident in the corresponding a_{r}, while the second enables partial information on the disconnected components of a_{r} for arbitrarily large order r to be obtained from explicit knowledge of only the first few c_{s} coefficients.

As will be made clear shortly, the structural simplicity that is exhibited by the exponential expansion has its origin in the well-known fact that the heat kernel on a product manifold factors into a tensor product of the heat kernels on the individual factor manifolds. Therefore, we begin by recording some facts concerning differential geometry and differential operators on product manifolds. The reader can think of the operator we are constructing as the Hamiltonian of a system of n distinguishable quantum particles, where the wave functions of the individual particles may have spinors, isovectors, etc. as values, and the background space may be curved.

Let n ∈ N and, for each j ∈ {1, 2, …, n}, let M_{j} be a C^{∞} paracompact real finite-dimensional manifold, let E_{j} be a C^{∞} real or complex finite-dimensional vector bundle with base M_{j} , and let E_{j}^{∗} be the corresponding dual bundle. Let M = ∏_{j=1}^{n} M_{j} be the product manifold, let p_{j}: M → M_{j} be the canonical projection onto the jth factor, let p_{j}^{∗}E_{j} and p_{j}^{∗}E_{j}^{∗} be the corresponding induced vector bundles with base M, and let E = ⨂_{j=1}^{n} p_{j}^{∗}E_{j} and E^{∗} = ⨂_{j=1}^{n} p_{j}^{∗}E_{j}^{∗} be the corresponding tensor-product bundles. The following are consequences of this product-manifold structure:

• There exist C^{∞}(M_{j})-linear mappings p_{j}^{∗}: Γ^{∞} (T (M_{j})) → Γ^{∞}(T (M)) and p_{j}^{∗}: Γ^{∞} (T^{∗}(M_{j})) → Γ^{∞}(T^{∗}(M)) such that, for all ω ∈ Γ^{∞} (T^{∗}(M_{i})) and

X ∈ Γ^{∞}(T (M_{j})),
⟨p_{i}^{∗}(ω), p_{j}^{∗}(X)⟩ = δ_{ij} p_{j}^{∗} (⟨ω, X⟩) .

• There exist C^{∞}(M_{j})-linear mappings p_{j}^{∗}: Γ^{∞}(E_{j}) → Γ^{∞}(p_{j}^{∗}E_{j}) and p_{j}^{∗}: Γ^{∞}(E_{j}^{∗}) → Γ^{∞}(p_{j}^{∗}E_{j}^{∗}) such that, for all v ∈ Γ^{∞}(E_{j}^{∗}) and u ∈ Γ^{∞}(E_{j}),

⟨p_{j}^{∗}(v), p_{j}^{∗}(u)⟩ = p_{j}^{∗} (⟨v, u⟩) .

The domain of the pull-back mapping p_{j}^{∗} may be extended to the space of smooth sections of any tensor-product bundle formed from T (M_{j}), T^{∗}(M_{j}), E_{j} , or E_{j}^{∗} by defining its action to be that of the C^{∞}(M_{j})-linear tensor-product mapping. This extended mapping p_{j}^{∗} commutes with contraction.

If we now introduce further structure by assuming that, for each j ∈ {1, 2, …, n}, T^{∗}(M_{j}) and E_{j} are equipped with C^{∞} connections ∇: Γ^{∞} (T^{∗}(M_{j})) → Γ^{∞} (T^{∗}(M_{j}) ⊗ T^{∗}(M_{j})) and ∇: Γ^{∞}(E_{j}) → Γ^{∞} (E_{j} ⊗ T^{∗}(M_{j})), then the mapping p_{j}^{∗} may be used to define natural C^{∞} induced connections on T^{∗}(M) and on p_{j}^{∗}E_{j} as follows:

• ∇: Γ^{∞}(T^{∗}(M)) → Γ^{∞} (T^{∗}(M) ⊗ T^{∗}(M)) is defined by requiring that the Leibniz formula is satisfied and that the following diagram commutes for each j ∈ {1, 2, …, n}:

Γ^{∞} (T^{∗}(M_{j})) ⟶^{∇} Γ^{∞} (T^{∗}(M_{j}) ⊗ T^{∗}(M_{j}))
p_{j}^{∗}↓  p_{j}^{∗}↓
Γ^{∞}(T^{∗}(M)) ⟶^{∇} Γ^{∞} (T^{∗}(M) ⊗ T^{∗}(M)) .

• ∇: Γ^{∞} (p_{j}^{∗}E_{j}) → Γ^{∞} (p_{j}^{∗}E_{j} ⊗ T^{∗}(M)) is defined for each j ∈ {1, 2, …, n} by requiring that the Leibniz formula is satisfied and that the following diagram commutes:

Γ^{∞}(E_{j}) ⟶^{∇} Γ^{∞} (E_{j} ⊗ T^{∗}(M_{j}))
p_{j}^{∗}↓  p_{j}^{∗}↓
Γ^{∞}(p_{j}^{∗}E_{j}) ⟶^{∇} Γ^{∞} (p_{j}^{∗}E_{j} ⊗ T^{∗}(M)) .

For each j ∈ {1, 2, …, n}, T (M_{j}) and E_{j}^{∗} may be assigned connections dual to those on T^{∗}(M_{j}) and E_{j} , respectively. The corresponding induced connections on T (M) and p_{j}^{∗}E_{j}^{∗}, which are defined analogously to the above, are dual to those on T^{∗}(M) and p_{j}^{∗}E_{j} . It will be assumed that any tensor-product bundle inherits the corresponding tensor-product connection. Since the action of p_{j}^{∗} on smooth sections of such a bundle is that of the corresponding tensor-product mapping, it follows that p_{j}^{∗} commutes with ∇.

As a consequence of these definitions of induced connections, the differential geometry of (E, M) is simply related to that of {(E_{j}, M_{j})}. This is perhaps most clearly illustrated by the following readily established equations, which show that the fundamental geometrical objects (torsion, manifold curvature, and bundle curvature) on (E, M) are direct sums of the pull-backs of the corresponding objects on {(E_{j}, M_{j})}:

T = ⨁_{j=1}^{n} p_{j}^{∗} T,  (6)

R = ⨁_{j=1}^{n} p_{j}^{∗} R,  (7)

and

R = ⨁_{j=1}^{n} I_{1} ⊗ ⋯ ⊗ I_{j−1} ⊗ p_{j}^{∗} R ⊗ I_{j+1} ⊗ ⋯ ⊗ I_{n} ,  (8)

where I_{j} denotes the identity endomorphism of p_{j}^{∗}E_{j} . Since ∇ commutes with p_{j}^{∗}, this direct sum structure also holds for the multiple covariant derivatives ∇^{m}T, ∇^{m}R, and ∇^{m}R, for each m ∈ N.

Let us now return to the fact that the heat kernel on a product manifold is a tensor product of the heat kernels on the individual factor manifolds. For each j ∈ {1, 2, …, n}, let A_{j}: Γ^{∞}(E_{j}) → Γ^{∞}(E_{j}) be a strongly elliptic secondorder differential operator with scalar principal symbol; each such operator may be represented as A_{j} = ∑_{r=0}^{2} A_{r}∇^{2−r}, where, for each r ∈ {0, 1, 2}, the coefficient tensor A_{r} (whose j dependence is suppressed) is in Γ^{∞} (End(E_{j}) ⊗ S (⊗^{2−r}T (M_{j}))) and contraction with ∇^{2−r} is understood. Given the product-manifold structure and the definitions of the induced connections, there is a natural strongly elliptic second-order differential operator A: Γ^{∞}(E) → Γ^{∞}(E) with scalar principal symbol that may be formed from the {A_{j}}, namely A = ∑_{r=0}^{2} A_{r}∇^{2−r}, where, for each r ∈ {0, 1, 2}, the coefficient tensor A_{r} ∈ Γ^{∞} (End(E) ⊗ S (⊗^{2−r}T (M))) is defined as the direct sum of the pull-backs of the corresponding coefficient tensors of the {A_{j}}:

A_{r} = ⨁_{j=1}^{n} I_{1} ⊗ ⋯ ⊗ I_{j−1} ⊗ p_{j}^{∗} A_{r} ⊗ I_{j+1} ⊗ ⋯ ⊗ I_{n} .  (9)

Since the operators under study are strongly elliptic with scalar principal symbol, this definition reduces for the leading-order coefficient to

A_{0} = ⨁_{j=1}^{n} p_{j}^{∗} A_{0} ,  (10)

which, in turn, implies the corresponding relationship

A^{0} = ⨁_{j=1}^{n} p_{j}^{∗} A^{0}  (11)

for its inverse. Furthermore, since ∇ commutes with p_{j}^{∗}, the direct-sum structure also holds for the multiple covariant derivatives ∇^{m}A_{r} for each m ∈ N.

As a consequence of this definition, the action of A: Γ^{∞}(E) → Γ^{∞}(E) on a tensor-product section of the form p_{1}^{∗}(u_{1}) ⊗ ⋯ ⊗ p_{n}^{∗}(u_{n}), where u_{j} ∈ Γ^{∞}(E_{j}) for each j ∈ {1, 2, …, n}, is as follows:

A (p_{1}^{∗}(u_{1}) ⊗ ⋯ ⊗ p_{n}^{∗}(u_{n})) = ∑_{j=1}^{n} p_{1}^{∗}(u_{1}) ⊗ ⋯ ⊗ p_{j−1}^{∗}(u_{j−1})
⊗ p_{j}^{∗} (A_{j}(u_{j})) ⊗ p_{j+1}^{∗}(u_{j+1}) ⊗ ⋯ ⊗ p_{n}^{∗}(u_{n}). (12)

Thus, if we denote by K_{j}(x_{j}, y_{j}, t) the integral kernel of the heat operator e^{−tA_{j}} associated to A_{j} , which satisfies

(\frac{∂}{∂t} + A_{j}) K_{j}(x_{j}, y_{j}, t) = 0  (13)

for t > 0, and effectively replace each u_{j} in (12) by K_{j} , then it readily follows that

K = p_{1}^{∗}(K_{1}) ⊗ ⋯ ⊗ p_{n}^{∗}(K_{n})  (14)

is the integral kernel of the heat operator e^{−tA} associated to A. This is a more precise statement of the fact that the heat kernel on a product manifold is a tensor product of the heat kernels on the individual factor manifolds.

If the conventional expansion (1) is substituted into (14), there results the following identity for the corresponding coefficients:

a_{r} = ∑_{s_{1}≥0,...,s_{n}≥0}^{s_{1}+···+s_{n}=r} p_{1}^{∗} (a_{s_{1}}) ⊗ ⋯ ⊗ p_{n}^{∗} (a_{s_{n}})  (15)

for each order r ≥ 0. On the other hand, substitution of the exponential expansion (2) results in the corresponding identity

c_{r} = ∑_{j=1}^{n} I_{1} ⊗ ⋯ ⊗ I_{j−1} ⊗ p_{j}^{∗} (c_{r}) ⊗ I_{j+1} ⊗ ⋯ ⊗ I_{n}  (16)

for each order r ≥ 1. The identity (15) suffers from the disadvantage that it mixes coefficients of terms of different orders; in contrast, the identity (16) does not. Clearly, the exponential expansion (2) is the more natural to use if it is desired to capture the content of the functional identity (14) order-by-order in the asymptotic expansion.

It was noted earlier that, like the corresponding a_{r}, each c_{r} is an invariant that is formed from the noncommutative associative algebra generated by {∇^{m}T, ∇^{m}R, ∇^{m}R, ∇^{m}A_{r}, A^{0}} by means of tensor contraction. On a product manifold, however, each of these generators has a direct-sum structure, as illustrated in equations (6)– (11), so there must be severe restrictions on the form of this invariant if it is to satisfy the identity (16); in particular, the product of two nontrivial invariants satisfying (16) will violate (16), because of the cross terms. This is the essence of the connectedness property. In order to ascertain the nature of these restrictions in rigorous generality, we will employ the representation theory of Lie algebras [16, 30].

Let U be a free associative algebra generated by the finite set of indeterminates {x_{i}} and let L be the smallest subalgebra of U_{L} which contains all the x_{i}, where U_{L} is the Lie algebra of U with product [xy] = xy −yx. Then L is a free Lie algebra generated by {x_{i}} and U is the universal enveloping algebra of L. There therefore exists a unique homomorphism δ: U → U ⊗ U, called the diagonal mapping, such that, for each i,

δ(x_{i}) = x_{i} ⊗ 1 + 1 ⊗ x_{i} .  (17)

In fact, such behavior under δ characterizes the Lie elements, as is made clear by the following theorem of Friedrichs: If x ∈ U then x ∈ L if and only if

δ(x) = x ⊗ 1 + 1 ⊗ x.  (18)

In order to apply these results to the problem at hand, we identify {∇^{m}T, ∇^{m}R, ∇^{m}R, ∇^{m}A_{r}, A^{0}} with {x_{i}} and set

δ(x_{i}) = p_{1}^{∗}(x_{i}) ⊗ I_{2} + I_{1} ⊗ p_{2}^{∗}(x_{i}).  (19)

Then the Friedrichs theorem informs us that the only elements of the noncommutative algebra generated by {∇^{m}T, ∇^{m}R, ∇^{m}R, ∇^{m}A_{r}, A^{0}} that have the direct-sum structure on a product manifold are the Lie elements.

Each c_{r} may therefore be viewed as a linear combination of monomial invariants, each of which has the form

I = C(L_{1} ⋯ L_{k}),  (20)

where L_{1}, …, L_{k} are elements of the Lie algebra generated by {∇^{m}T, ∇^{m}R, ∇^{m}R, ∇^{m}A_{r}, A^{0}} and C denotes a tensor contraction operation. Since each L_{i} factor has the direct-sum structure on a product manifold, there must be severe restrictions on the allowed form of the tensor contraction C if I is to have the behavior exhibited in (16). In fact, C must link together all the L_{i} factors, so that I has the structure of a connected graph in which the L_{i} factors are represented as vertices and the contravariant-covariant tensor contractions are represented as edges. Thus, each c_{r} is expressible as a linear combination of monomial invariants, each of which is a connected product of elements of the Lie algebra generated by {∇^{m}T, ∇^{m}R, ∇^{m}R, ∇^{m}A_{r}, A^{0}}. This is our principal conclusion.

Let us illustrate this conclusion by reference to the covariant Schrödinger operator A = −Δ + V (where Δ is the Laplace–Beltrami operator on M). Although the invariant R_{αβγδ}R^{αβγδ}R_{μν}R^{μν}, for example, is permitted in a_{4} , this invariant cannot appear in c_{4} , since its tensor contractions do not connect together all the factors, and it therefore does not correspond to a connected graph. Likewise, the only way in which a product involving only V and V_{;μ}^{μ} can appear in c_{4} is in the form of the Lie product [V, [V, V_{;μ}^{μ}]].

With the recognition that each c_{r} has a connected-graph structure comes the realization that each a_{r} may be decomposed into a sum of connected and disconnected components. Indeed, (4) may be rewritten as

a_{r} = ∑_{q=1}^{r} ∑_{1≤s_{1}≤···≤s_{q}}^{s_{1}+···+s_{q}=r} c_{s_{1},...,s_{q}} ,  (21)

where

c_{s_{1},...,s_{q}} = \frac{1}{q!} ∑_{π∈P (s_{1},...,s_{q})} c_{s_{π(1)}} ⋯ c_{s_{π(q)}}  (22)

and P (s_{1}, …, s_{q}) is the set of distinct permutations of the ordered multi-index (s_{1}, …, s_{q}), and it can be shown that (21) is a direct-sum decomposition of a_{r}, with c_{r} being the connected component and each c_{s_{1},...,s_{q}} being a disconnected component of order q ≥ 2. In particular, it follows from this decomposition of a_{r} that explicit knowledge of only the first few c_{s} coefficients may, through (22), be used to determine some of the highly disconnected components of a_{r} for arbitrarily large order r.

Throughout this paper we have restricted our attention to the diagonal values of the heat kernel. However, similar properties hold for the exponential expansion of the off-diagonal heat kernel. Full proofs of these results will be published elsewhere. However, we shall close by demonstrating by direct calculation that the first nontrivial term in the off-diagonal exponential expansion for a vector-valued Schrödinger system does have the Lie-connected structure prescribed by our theorem. We are grateful to Tom Osborn for discussions during the conference, in which this exploratory calculation was made.

According to [25] (with suitable notational changes) the heat kernel for the operator A = −Δ + V (x), where x ∈ R^{d} and V (x) is matrix-valued, has the asymptotic expansion

K(x, y, t) ∼ K_{0}(x, y, t) ∑_{r=0}^{∞} a_{r}(x, y) t^{r}  (23)

as t → 0^{+}, where K_{0} is the free heat kernel (i.e., that with V ≡ 0) and

a_{0}(x, y) = I, a_{1}(x, y) = − ∫_{0}^{1} dξ_{1} V (ξ̂_{1}),

a_{2}(x, y) = ∫_{0}^{1} dξ_{1} ∫_{0}^{ξ_{1}} dξ_{2} V (ξ̂_{2})V (ξ̂_{1}) − ∫_{0}^{1} dξ_{1} ξ_{1}(1 − ξ_{1}) ΔV (ξ̂_{1}). (24)

Here

ξ̂_{i} = (1 − ξ_{i})x + ξ_{i} y  (25)

is the parametrized line segment from x to y.

In analogy with (2) and (5), we write

K(x, y, t) ∼ K_{0}(x, y, t) exp [∑_{r=1}^{∞} c_{r}(x, y) t^{r}] ,  (26)

and

c_{1}(x, y) = a_{1}(x, y),  c_{2}(x, y) = a_{2}(x, y) − \frac{1}{2} (a_{1}(x, y))^{2} ,  etc. (27)

From (27) and (24) one gets

c_{2}(x, y) + ∫_{0}^{1} dξ_{1} ξ_{1}(1 − ξ_{1}) ΔV (ξ̂_{1})

= ∫_{0}^{1} dξ_{1} ∫_{0}^{ξ_{1}} dξ_{2} V (ξ̂_{2})V (ξ̂_{1}) − \frac{1}{2} ∫_{0}^{1} dξ_{1} V (ξ̂_{1}) ∫_{0}^{1} dξ_{2} V (ξ̂_{2})

= \frac{1}{2} ∫_{0}^{1} dξ_{1} ∫_{0}^{ξ_{1}} dξ_{2} [V (ξ̂_{2}), V (ξ̂_{1})]

+ \frac{1}{2} ∫_{0}^{1} dξ_{1} ∫_{0}^{ξ_{1}} dξ_{2} V (ξ̂_{2})V (ξ̂_{1}) − \frac{1}{2} ∫_{0}^{1} dξ_{1} ∫_{ξ_{1}}^{1} dξ_{2} V (ξ̂_{1})V (ξ̂_{2})

= \frac{1}{2} ∫_{0}^{1} dξ_{1} ∫_{0}^{ξ_{1}} dξ_{2} [V (ξ̂_{2}), V (ξ̂_{1})]

+ \frac{1}{2} ∫_{0}^{1} dξ_{1} ∫_{0}^{ξ_{1}} dξ_{2} V (ξ̂_{2})V (ξ̂_{1}) − \frac{1}{2} ∫_{0}^{1} dξ_{2} ∫_{0}^{ξ_{2}} dξ_{1} V (ξ̂_{1})V (ξ̂_{2}).

Since the last two terms cancel upon reindexing, one has

c_{2}(x, y) = \frac{1}{2} ∫_{0}^{1} dξ_{1} ∫_{0}^{ξ_{1}} dξ_{2} [V (ξ̂_{2}), V (ξ̂_{1})] − ∫_{0}^{1} dξ_{1} ξ_{1}(1 − ξ_{1}) ΔV (ξ̂_{1}), (28)

an expression of the expected type.

Clearly, this calculation in the reverse direction would become increasingly tedious and pointless at higher orders. An efficient algorithm for directly calculating the c_{r}(x, y) needs to be discovered.

G. Kennedy’s attendance was supported in part by a Carnegie Trust Travel Grant.

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