Feynman diagrams and Wick products associated with q-Fock space

Abstract

It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute the q-Wick products and usual operator products in terms of each other.

1. Introduction

A recurrent theme in noncommutative analysis is that one may use graphs to efficiently index the terms in complicated sums. One of the first to recognize this principle was Cayley (1), who introduced rooted trees in order to label differentials (see ref. 2 for additional examples). Currently, the best known example of graph-theoretic indexing may be found in perturbative quantum field theory. In this context one uses Feynman diagrams to index summands that arise when one evaluates the expectations of products of jointly Gaussian random variables (see refs. 3 and 4).

The random variables of quantum field theory correspond to certain self-adjoint operators on symmetric or antisymmetric Fock spaces (see refs. 3 and 5). In 1991, Bożejko and Speicher (6) introduced a remarkable q version of the Fock space, which for q = 1, –1, and 0 coincides with the symmetric (Boson), antisymmetric (Fermion), and full (Voiculescu) Fock spaces (some of these ideas had been considered in ref. 7; see also ref. 8). Bożejko and Speicher's q versions of stochastic processes and second quantization have attracted the attention of a large number of researchers (see refs. 9–11).

In ref. 8, Bożejko et al. introduced the q analogs of the Wick product. We show that some of the basic combinatorial calculations involving Wick products of Gaussian random variables have natural q versions. In particular, we use Feynman diagrams to express the Wick products in terms of the usual operator products and vice versa.

We will explore q forms of the Hopf algebraic theory of Kreimer (12) and Connes and Kreimer (13) in a subsequent paper.

2. q-Fock Spaces and Feynman Diagrams

We begin by recalling the Bożejko and Speicher (6) theory. Let H₀ be a real Hilbert space and let H be its complexification. We write

for the algebraic tensor product, and given –1 ≤ q ≤ 1, we define a hermitian form on the algebraic sum

where Ω is taken to be a unit vector, by letting

and where

(we use # to indicate cardinality). We will generally delete the subscript q in the hermitian form. The q-Fock space F_q(H) is the completion of this pre-Hilbert space. If q = 1or q = –1, we must first divide out by the null space, and we then obtain the usual symmetric and antisymmetric Fock spaces. If q = 0 we obtain the full Fock space. Unless otherwise indicated, we restrict our attention to the case –1 < q < 1. We let and

By an elementary tensor we mean an element in of the form .

For each f ∈ H₀, the creation and annihilation operators a^–(f) and a^–(f) are defined on (H) by

[1]

and

[2]

If q < 1, a^–(f) and a^–(f) extend to bounded operators on F_q(H) satisfying a^–(f) = a^–(f)*. They satisfy the commutation relation

We refer to the operators

as “q Gaussians.” For q = 1, these may be identified with jointly Gaussian random variables (see theorem I.11 in ref. 5). We define Γ_q(H₀) to be the von Neumann algebra on F_q(H) generated by these operators. The vector Ω is separating and cyclic for Γ_q(H₀). We let

be the corresponding state on Γ_q(H₀). In particular, if ξ = φ(f) and η = φ(g) for f, g ∈ H₀, then we have the covariance

(we recall that H₀ is a real Hilbert space).

We wish to compute the “multivariable moments”

of q Gaussians ξ_i = φ(f_i), 1 ≤ i ≤ m. As in the classical case, these are determined by polynomials of the “covariances” .

We begin by letting a^–1(f) = a^–(f) and a^–1(f) = a^–(f). Our first task is to compute expressions of the form

for sequences ε = (ε(1),..., ε(2n)) ∈ I(2n) = {1, –1}²ⁿ.

Given ε ∈ I(2n), we define

A simple induction shows that if σ₁ > 0, σ₂,..., σ_2n ≥ 0, then

[3]

and otherwise a^ε(k)(f_k)... a^ε(2n)(f_n)Ω = 0. We say that ε ∈ I(2n) is a Catalan sequence if σ_2n > 0, σ_2n–1 ≥ 0,..., σ₁ = 0, and we let C(2n) be the set of such sequences. It is evident that m(ε) = 0 unless ε ∈ C(2n). We may associate a Dyck path (see ref. 15, p. 221) with each ε ∈ C(2n). For example, if ε = (–1, –1, 1, –1, –1, 1, 1, 1) ∈ C(8), then the corresponding Dyck path is given by

where each ε(k) is the slope of the corresponding line segment. If ε ∈ C(2n), we may evaluate m(ε) in terms of “Feynman diagrams” on [2n] = {1,..., 2n} (see refs. 3 and 4 for this terminology).

Given a finite linearly ordered set S, a Feynman diagram γ on S is a partition of S into one- and two-element sets. It will be convenient to regard γ as a set of ordered pairs {(i₁, j₁),..., (i_p, j_p)} with i_k < j_k and i_k ≠ i_l and j_k ≠ j_l for k ≠ l, and we refer to the unpaired indices as “singletons.” With this notation we will assume that

[4]

(the j_k will generally be out of order). We write S^– (respectively, S^–) for the j ∈ S with (k, j) ∈ γ for some k [respectively, the i ∈ S such that (i, k) ∈ γ for some i]. We refer to the elements of S^– and S^– as creators and annihilators, respectively. We may specify a Feynman diagram with a simple graph of the following form

This corresponds to the Feynman diagram γ = {(1, 3), (2, 6), (4, 9), (8, 10)} on [10]. We let F(S) denote the set of all Feynman diagrams on S, and we let F(n) = F([n]).

We note that more general partitions and their crossings are analyzed by using a succession of semicircles to link the elements of an equivalence class (see ref. 14).

We call the elements in S the “vertices” of the diagram. We say that a pair (k, l) ∈ γ is a “left crossing” for (i, j) if k < i < l < j, and we define c_l(i, j) to be the number of such left crossings. We refer to c(γ) = ∑_(i,j)∈γ c_l(i, j) as the “crossing number” of γ (Biane calls this the “restricted crossing number” in ref. 14). The total left crossings can be found by counting the intersections in the corresponding graph. In the diagram shown above, c_l(1, 3) = 0 and c_l(2, 6) = c_l(4, 9) = c_l(8, 10) = 1, and thus

The general result is evident if one notes that an intersection in the graph will occur between an ascending line for one pair (k, l) ∈ γ and a descending line for another (i, j) ∈ γ, which in turn will correspond to the left crossing (k, l) of (i, j). Similarly, c(γ) = ∑_(i,j)∈γ c_r(i, j), where c_r(i, j) is the number of right crossings i < k < j < l where (k, l) ∈ γ.

If (i, j) ∈ γ, we define the “gap” g(i, j) to be the number of k with i < k < j, and we let a(i, j) = g(i, j) – c_l(i, j) (we will not need the right version of this). We define g(γ) = ∑_(i,j)∈γ g(i, j) and a(γ) = ∑_(i,j)∈γ a(i, j) = g(γ) – c(γ). Given (i, j) ∈ γ, it is evident that c_l(i, j) + c_r(i, j) ≤ g(i, j), and thus

[5]

For some purposes, it is also useful to count “degenerate crossings.” These are the triples i < k < j, where k is not paired and (i, j) ∈ γ. We let d(γ) be the number of such triples in γ, and we define the “total crossing number” to be tc(γ) = c(γ) + d(γ).

A Feynman diagram γ on S is complete if the there are no singletons (in which case S must have an even number of elements), and we let F_c(S) be the collection of all such diagrams. Given ε ∈ C(2n), we say that a complete Feynman diagram,

on [2n] is compatible with ε if ε(i_k) = –1 and ε(j_k) = 1. We let F_c(ε) be the set of all such Feynman diagrams on S. It is easy to see that F_c(ε) is nonempty. For example, we may fix a vertex of maximum height and then pair the decending and ascending edges adjacent to that vertex. In the diagram shown above we begin by placing the pair (5, 6) in γ. Eliminating the two line segments and rejoinging the graph, we can continue by induction to appropriately pair all the ascending edges with descending edges. Conversely, each complete Feynman diagram γ on [2n] is compatible with the unique sequence ε ∈ C(2n), defined by letting ε(k) = 1 if k ∈ [2n]^– and ε(k) = –1 if k ∈ [2n]^–.

Given q-Gaussian random variables ξ_i = φ(f_i) (i = 1,..., n) and a Feynman diagram γ on [n], we let

where γ = {(i_k, j_k): k = 1,..., p}, and h₁ < h₂ <... < h_r are the γ singletons.

The following is due to Bożejko and Speicher (see proposition 2 in ref. 6). We have included an alternative proof for the convenience of the reader.

Theorem 2.1. For any ε ∈ C(2n),

Proof: Given γ ∈ F_c(ε), we will assume, as before, that γ = {(i_k, j_k)}, where i₁ <... < i_n. We define a sequence of elementary tensors,

as follows. Let us suppose that i_n < i_n + 1 <... < j_n ≤ 2n. We define

From Eq. 2,

We define to be a particular elementary tensor summand in this expression:

If i_n < k < j_n, then k ∈ [2n]⁺, i.e., there is an h with (h, k) ∈ γ. Because h < i_n, it follows that (h, k) is a left crossing for (i_n, j_n), and we see that there are exactly c(i_n, j_n) = j_n – i_n – 1 terms between i_n and j_n. We conclude that

Let us suppose that we have defined elementary tensors in such a manner that i_p < k ≤ i_p–1, and none of the factors occur in . If k > i_p + 1, we let

On the other hand, if k = i_p + 1, let us suppose that

for some scalar α. We define

which is one of the elementary tensor summands of . Each k_t lies in [2n]⁺\{j_p–1,..., j_n} and k_t < j_p; hence if k_t = j_h, then h < p and therefore i_h < i_p. It follows that (i_h, k_t) is a left crossing for (i_p, j_p). Because this holds for each t, we see that r = c_ℓ(i_p, j_p). Continuing in this manner, .

Because it is evident that every nonzero elementary tensor summand in

corresponds to a unique Feynman diagram in F_c(ε),

and we are done.

Corollary 2.1 (q-Wick Theorem). For any q-Gaussian random variables ξ_i, we have

and on the other hand,

Proof: We have that

and thus the formula follows from the theorem. The result for odd moments is immediate from Eq. 3.

3. q-Wick Products

Following ref. 8, we define the q-Wick product for f₁,..., f_n ∈ H by

[6]

where the sum is taken over all families I = {i₁,..., i_k} and J = {j₁,..., j_l} where i₁ <... < i_k, j₁ <... < j_l, and , and we let ι(I, J) = #({(p, q): i_p > j_q}. This operator is characterized by the recursion

[7]

(see proof of proposition 2.7 in ref. 8). It follows from a simple induction on Eq. 7 that W(f₁,... f_n) is a polynomial of the noncommuting operators ξ_i = φ(f_i), (1 ≤ i ≤ n), and we will use the usual Wick product notation

Because Ω is separating for Γ_q(H₀), u =: ξ₁... ξ_n: is the unique operator in Γ_q(H₀) satisfying

We may linearly extend this to define:p(ξ₁,..., ξ_n): for any polynomial p(ξ₁,..., ξ_n).

The following provides an explicit (nonrecursive) expression for the q-Wick product analogous to the “classical formula” for q = 1 (see theorem 3.4 in ref. 4). We let #(γ) denote the number of pairs in γ.

Theorem 3.1. For any q-Gaussian random variables ξ_i = φ(f_i) (i = 1,..., n)

[8]

Proof: F(1) contains only the empty Feynman diagram γ₀, and

On the other hand, F(2) = {γ₀, γ₁}, where γ₀ is empty and γ₁ = {(1, 2)}. We have

Let us suppose that we have proved Eq. 8 for n – 1 and n. If ξ₀ = φ(f₀), then applying the formula for n and n – 1 and the recurrence relation,

Each Feynman diagram γ = {(i_k, j_k)} ∈ F(n) trivially determines a Feynman diagram , for which ξ₀ν(γ) = ν(γ′). Because #(γ′) = #(γ) and a(γ′) = a(γ),

Each Feynman diagram δ ∈ F([n]{l}) determines a Feynman diagram for which

It is evident that #(δ′) = #(δ) + 1. Because c_δ′,ℓ(0, ℓ) = 0, a_δ′(0, l) = l – 1. If (i, j) ∈ δ, then we may consider three cases. If l < i, or j < l, then it is evident that a_δ′(i, j) = a_δ(i, j). If i < l < j, then g_δ′(i, j) = g_δ(i, j) + 1. On the other hand, 0 < i < l < j introduces another left crossing when we regard (i, j) as an element of δ′, and thus c_δ′(i, j) = c_δ(i, j) + 1. It follows that a(δ′) = l – 1 + a(δ) and

A Feynman diagram θ in has the form γ′ if and only if 0 is a singleton in θ, and the form δ′ if (0, l) ∈ θ. Thus

[9]

and we are done.

Conversely, we may express products ξ₁... ξ_n in terms of q-Wick products. For this purpose we need to generalize the q-Wick theorem to products of q-Wick products. Given q-Gaussian random variables {ξ_p,k} with 1 ≤ p ≤ t and 1 ≤ k ≤ n_p, we may regard the index set S = {(p, k)} as partitioned by the first integer, and we refer to each partition as a “block.” We let S_lex denote S with the lexicographic ordering

Theorem 3.2. Suppose that we are given q-Gaussian random variables {ξ_p,k} with 1 ≤ p ≤ t and 1 ≤ k ≤ n_p. Then if Y_p =: ξ_p,1... ξ_p,np:, we have

where the sum is taken over all complete Feynman diagrams γ on S_lex that do not link vertices within blocks.

Proof: A typical summand of Y₁... Y_t has the form

where for each p, j₁ < j₂ <... and i₁ < i₂ <....

We may use Theorem 2.1 to compute 〈uΩ, Ω〉 provided we reorder the index set. We let S_u denote S with the total ordering (p, k) < (p′, k′) if p < p′, and

[10]

If we let ε(p, i_g) = 1 and ε(p, j_h) = –1, then

where ι(I, J) = ∑ι(I_k, J_k). It should be noted that if γ ∈ F_c(ε), then γ will not link elements of a block, because in the definition of ν(γ) a creator is always paired with an annihilator on its left.

We may use a sequence of ι(I, J) transpositions of the index set S to transform S_u into S_lex. Retaining the same ordered pairs, each Feynman diagram γ on S with a given total ordering is mapped to a Feynman diagram γ′ on the reordered set. It is evident that ν(γ) = ν(γ′), but in general the number of crossings will change.

If S_u ≠ S_lex, then i_r(p) > j₁ for some p. Our first step will be the transition from

to

Continuing in this manner, a series of ι(I, J) transpositions will give us a chain of Feynman diagrams γ₁ → γ₂ →... → γ_ι(I,J) on the permuted sets, which will return us to the lexicographic ordering.

At each stage we will have an adjacent “disordered” pair (p, i_k), (p, j_l) with i_k > j_l, and we perform the transposition

Let us consider the crossings in the corresponding Feynman diagrams γ and γ′. Suppose that a < b < c < d is a crossing in either γ or γ′. If none of these four vertices coincides with i_k or j_l, the crossing will be left invariant under the transpositions γ ↔ γ′. We have three other cases [remember that c, d and (p, j_l) are creators, and the terms (p, i_k), (p, j_l) are adjacent]:

In the first and second cases, the crossing a < b < c < d will remain unaffected under these transpositions. From our construction, the third case will occur precisely when the noncrossing sequence a < (p, i_k) < (p, j_ℓ) < d occurs in γ with i_k > j_l, and we obtain γ′ by the transposition. It follows that c(γ′) = c(γ) + 1, and thus q^c(γ)q^h = q^c(γ′)q^h–1.

Starting with a Feynman diagram γ = γ₀ on S_u, we obtain a Feynman diagram γ′ = γ_ι(I,J) on S_lex with ν(γ)q^c(γ)q^ι(I,J) = ν(γ′)q^c(γ′). It is easy to see that all complete diagrams on S_lex that do not link elements within blocks arise in this fashion; hence taking the sum of terms u, we obtain the desired result.

Theorem 3.3. Let Y_i =: ξ_i1... ξ_ini:. Then

[11]

where the sum is taken over all Feynman diagrams γ on S_lex that do not link vertices within blocks.

Proof: Let A = Y₁... Y_t and B = ∑:ν(γ): q^tc(γ). As in the proof of theorem 3.15 in ref. 4, it suffices to show that for any Wick product W =: η₁... η_u:, . From Theorem 3.2,

where we sum over all complete Feynman diagrams δ on the partitioned ordered set

(where the semicolons separate blocks), which do not link vertices in the same block. Each such diagram determines a (generally incomplete) diagram on S_lex with the same property. Because δ is complete, the singletons (p₁, k₁) <... < (p_u, k_u) in γ are linked with elements of [u] = {1,..., u}. It follows that

where θ is a complete Feynman diagram on

that does not link vertices in the two blocks.

On the other hand, given a Feynman diagram γ on S that does not link elements in any of the blocks,

and thus

From Theorem 3.2,

where G is the set of all complete Feynman diagrams θ on T₀ that do not link vertices of the two blocks. Given such a θ, is a generic complete Feynman diagram on T extending γ, which does not link internal vertices. It is evident that

because forming the term will “hide” precisely d(γ) intersections arriving from the pairs in θ. It follows that

where we sum over nonlinking extensions δ of γ, and thus

The one-variable case of the following result was proved by Anshelevich in remark 6.15 in ref. 11.

Corollary 3.1. Given ξ_j = φ(f_j) as above, we have

4. The Free Case (q = 0) and Noncrossing Diagrams

We say that a Feynman diagram γ on a finite totally ordered set S is noncrossing if c(γ) = 0, strongly noncrossing if tc(γ) = 0, and gap-free if g(γ) = 0. We let NC(S), SNC(S), and GF(S) denote the corresponding diagrams on S. It is evident that

If S = [n], we will simply write NC(n), etc.

If q = 0, we have the commutation relation a^–(f)a^–(g) = 〈 f, g〉 I. The convention of ref. 6 is that q^c = 0 for c ≠ 0, and q⁰ = 1. Thus we may drop terms in the 0-Wick theorem for which q is raised to a positive power:

Turning to Wick products, we delete terms with inversions in the definition of the free Wick product:

For the alternative formula (Eq. 8) we need only consider terms with a(γ) = g(γ) – c(γ) = 0. It follows from Eq. 5 that g(γ) = 0, and thus γ is gap-free. We conclude that

[12]

We have, for example, that

[13]

Turning to Eq. 11, we have

[14]

where the sum is over strongly noncrossing diagrams that do not link elements of a block. Thus, in particular,

[15]