Clifford Algebraic Approach to the De Donder–Weyl Hamiltonian Theory

Abstract

The Clifford algebraic formulation of the Duffin–Kemmer–Petiau (DKP) algebras is applied to recast the De Donder–Weyl Hamiltonian (DWH) theory as an algebraic description independent of the matrix representation of the DKP algebra. We show that the DWH equations for antisymmetric fields arise out of the action of the DKP algebra on certain invariant subspaces of the Clifford algebra which carry the representations of the fields. The matrix representation-free formula for the bracket associated with the DKP form of the DWH equations is also derived. This bracket satisfies a generalization of the standard properties of the Poisson bracket.

Appendix: Multilinear Endomorphisms in \(\mathrm {G}_n\)

In this Appendix we show that the splitting $W=V\oplus V^*$ of the vector space W into two complementary Lagrangian subspaces V and $V^*$ under the bilinear form, relation (1), in addition with the corresponding invariant projector (P), lead to a useful basis to expand the algebra $\mathrm {G}_n$. The convenience of this type of basis is that it makes explicit the appearance of the spaces of algebraic spinors [5, 6, 8, 16, 50] in $\mathrm {G}_n$. These spaces turn out to be the minimal left and right ideals of the algebra. We will also introduce the multi-index notation which is particularly useful when dealing with multilinear algebras.

1.1 The Projector Basis of the Split Form

Start with a dual basis $e_1,\ldots , e_n$ and $e^1,\ldots , e^n$ of vectors and covectors respectively of the complementary Lagrangian subspaces V and $V^*$. Set the invariant projectors as follows,

$$\begin{aligned} \Pi _{p}=(p\,!)^{-1}\sum _{j_1=1}^{n}\cdots \sum _{j_p=1}^{n} {}^{j_1,\ldots ,j_p}\!P_{j_p,\ldots ,j_1}, \end{aligned}$$

(38)

where

$$\begin{aligned} {}^{j_1,\ldots ,j_p}\!P_{j_p,\ldots ,j_1}:= e^{j_1}\cdots e^{j_p}(P)e_{j_p}\cdots e_{j_1}, \end{aligned}$$

(39)

and $p=0,\ldots ,n$, $\dim V =n$, with the convention that $\Pi _{0}:=P= {{\mathcal {N}}}_{1}\cdots {{\mathcal {N}}}_{n}=e_{1}e^{1}\cdots e_{n}e^{n}$. Notice that the projectors (38) are not postulated but they are constructed from Clifford products of isotropic basis vectors. This construction is due to Schönberg [50].

A multi-index I of length k is a k-tuple of positive integers, say $i_1 i_2\ldots i_k$, from the set $\{1,2,\ldots ,\dim V\}$. We are going to denote them by the upper-case Roman letters and the associated indices will be denoted by the lower case letters. A multi-index I is said to have the length p if $k=p$ in the k-tuple that represents I and in that case, we write $|I|=p$ for the length of I. We will also use the summation convention for the multi-index.

Using the multi-index notation we rewrite Eqs. (38) and (39) accordingly,

$$\begin{aligned} \Pi _{|J|}=(|J|!)^{-1}\sum _{J} {}^{J}\!P_{\overleftarrow{J}}, \end{aligned}$$

(40)

where $J=j_{1}\ldots j_{|J|}$; $\overleftarrow{J}:=j_{|J|}\ldots j_1$ and $|\overleftarrow{J}|=|J|=p$. The length |K| of the elements can only runs from 0 to dim V in the $\mathrm {G}_n$ algebra.

As a consequence of the relations (2), (3), (4) and (6), the elements (39) satisfy the relation,

$$\begin{aligned} ({}^{j_1,\ldots ,j_p}\!P_{k_q,\ldots ,k_1})({}^{j'_{1},\ldots ,j'_{p'}}\!P_{k'_{q'},\ldots ,k'_{1}})= \delta _{q,p'}\delta ^{j'_{1},\ldots ,j'_{p'}}_{k_1,\ldots ,k_q}({}^{j_1,\ldots ,j_p}\!P_{k'_{q'},\ldots ,k'_{1}}) , \end{aligned}$$

(41)

where $\delta _{q,p'}\delta ^{j'_{1},\ldots ,j'_{p'}}_{k_1,\ldots ,k_q}$ denotes the generalized Kronecker deltas,

$$\begin{aligned} \delta _{q,p'} \delta ^{j'_{1},\ldots ,j'_{p'}}_{k_1,\ldots ,k_q} = \det \left( \begin{array}{cccc} \delta ^{j'_1}_{k_1}&{} \delta ^{j'_1}_{k_2}&{} \cdots &{} \delta ^{j'_{1'}}_{k_q} \\ \delta ^{j'_2}_{k_1} &{} \delta ^{j'_2}_{k_2} &{} \cdots &{} \delta ^{j'_{2'}}_{k_q} \\ \cdot &{} \cdot &{} \cdot &{} \cdot \\ \delta ^{j'_{p'}}_{k_1} &{} \delta ^{j'_{p'}}_{k_2} &{} \cdots &{} \delta ^{j'_{p'}}_{k_q} \end{array} \right) , \end{aligned}$$

or succinctly stated,

$$\begin{aligned} ({}^{J}\!P_{\overleftarrow{K}})({}^{J'}\!P_{\overleftarrow{K'}})=\delta _{|\overleftarrow{K}|,|J'|} \Delta ^{J'}_{K} ({}^{J}\!P_{\overleftarrow{K'}}), \qquad |\overleftarrow{K}|=q, \; |J'|=p', \end{aligned}$$

(42)

where $\Delta ^{J'}_{K}\equiv \delta ^{j'_{1},\ldots ,j'_{p'}}_{k_1,\ldots ,k_q}$.

Relation (41) or (42) shows that the set of $2^{2n}$ elements (39) of $\mathrm {G}_n$ with $p,q=0,\ldots ,n$ and $j_{1}<\cdots <j_{p}$, $k_{1}<\cdots <k_{q}$, $j, k=1,\ldots ,n$ are linearly independent and thus form a basis of $\mathrm {G}_n$, which has dimension $2^{2n}$.

In particular, the generators $(e_j)$ and $(e^j)$ can be retrieved from the basis elements ${}^{j_1,\ldots ,j_p}\!P_{k_q,\ldots ,k_1}$ by writing

$$\begin{aligned} e^j&=\sum _p (p!)^{-1}({}^{j,j_1,\ldots ,j_p}\!P_{ j_p,\ldots ,j_1})=\sum _{|J|=0}^{n} (|J|!)^{-1}\;({}^{j,J}\!P_{\overleftarrow{J}}), \\ e_j&=\sum _p (p!)^{-1}({}^{j_1,\ldots ,j_p}\!P_{j_p,\ldots ,j_1,j})=\sum _{|J|=0}^{n} (|J|!)^{-1}({}^{J}\!P_{\overleftarrow{J},j}), \end{aligned}$$

which are seen to satisfy the defining relations (2)–(4) of the $\mathrm {G}_n$ algebra according to multiplication (41) or (42). Any element of the $\mathrm {G}_n$ algebra can be written in the basis $(^{J} P_{\overleftarrow{K}})$ as follows:

$$\begin{aligned} \Lambda= & {} \sum _{p,q=0}^n (p!q!)^{-1} A_{j_1,\ldots ,j_p}{}^{k_1,\ldots ,k_q} ({}^{j_1,\ldots ,j_p}\!P_{k_q,\ldots ,k_1}) \nonumber \\= & {} \sum _{|J|,|K|=0}^{n} A_{J}{}^{K}({}^{J}\! P_{\overleftarrow{K}}), \end{aligned}$$

(43)

where we can recognize the coefficients $A_{j_1,\ldots ,j_p}{}^{k_1,\ldots ,k_q}$ as tensor components. Note that, by writing all the terms of the sum explicitly there will occur all types of tensors including the zeroth rank elements regarded as the scalars, the mixed tensors and finally the antisymmetric ones, either covariant or contravariant up to rank n. The great advantage of studying the $\mathrm {G}_n$ algebra in this basis is that by using the system of multilinear projectors $\Pi _{|J|}$ we can easily project on tensor spaces of different lengths within $\mathrm {G}_n$. One of the most important projectors is the minimal idempotent $\Pi _0=(P)$. It projects $\mathrm {G}_n$ onto its minimal left/right ideals. That is, take $\Lambda \in \mathrm {G}_n$ written in the form (43) and compute the product or right projection $\Lambda (\Pi _0)=\Lambda (P)$ using rules (41). The result is

$$\begin{aligned} \psi =\sum _{p=0}^n (p!)^{-1} A_{j_1,\ldots ,j_p} ({}^{j_1,\ldots ,j_p}\!P)=\sum _{|J|=0}^n (|J|!)^{-1}A_J({}^J \! P). \end{aligned}$$

Let us call the space of these elements $\mathrm {G}_n(P)$. Such space is isomorphic to the direct sum of the linear spaces of homogeneous forms of all orders up to $n=\dim V$. The set $\mathrm {G}_n(P)$ is a minimal left ideal of $\mathrm {G}_n$. Clearly, due to rules (41) one can easily check that the elements $\Lambda \in \mathrm {G}_n$ are endomorphisms of $\mathrm {G}_n(P)$. So $\mathrm {G}_n(P)$ is the space of spinors of the vector space $W=V\oplus V^{*}$ with which we started.

Upon left projection, the analogous arguments lead to the minimal right ideal $(P)\mathrm {G}_n$ which is the dual of $\mathrm {G}_n(P)$.

It is common in the literature to define these spinors as elements of the Grassmann algebra of the Lagrangian subspace $V^{*}$ of W [13, 41]. This Grassmann algebra turns out to be a Clifford module. The dual spinors belong to the Grassmann algebra of the complementary space V. These Clifford modules are isomorphic to the minimal left/right ideals of $\mathrm {G}_n$. These minimal ideals are known as algebraic spinor spaces [5, 6, 8, 46, 50].

By using the system of multilinear projectors $\Pi _p$, we see that the projection $\Pi _p\mathrm {G}_n(P)$ is isomorphic to the space of homogeneous forms of order (rank) p. By using the $(\Pi _p)$ we can also write an element of $\mathrm {G}_n(P)$ in a basis free form

$$\begin{aligned} \psi =\sum _{|J|=0}^n (\Pi _{|J|})\Lambda (P), \end{aligned}$$

where $\Lambda \in \text {G}_n$.

From the multiplication rule (41) the following useful algebraic properties can be obtained which are used throughout the text

$$\begin{aligned} \sum _{|J|=0}^{n} \Pi _{|J|}=1_{G_n}, \quad \Pi _{|J|}(\Pi _{|K|})=\delta _{|J|,|K|}(\Pi _{|J|}) \end{aligned}$$

and

$$\begin{aligned} \alpha (\Pi _{|J|})=(\Pi _{|J|+1})\alpha , \qquad \Pi _{|J|}(v)=v(\Pi _{|J|+1}). \end{aligned}$$

(44)

In plain words, $\Pi _{|J|}$ changes to $\Pi _{|J|+1}$ when it moves left past a covariant vector, and $\Pi _{|J|}$ changes to $\Pi _{|J|+1}$ when it moves right past a contravariant vector. We also assume that $\Pi _{-1}=0$ and $\Pi _{n+1}=0$.

1.2 Adjunction and Contraction

Two further operations in the algebra can be defined. The first operation is adjunction which is defined as follows [50],

$$\begin{aligned} (e_i)^{\dag }=(e_i,0)^{\dag }:=(0,g_{ij}e^j)=(0,{\widetilde{e}}_i)=g_{ij}(e^j)\equiv ({\widetilde{e}}_i) \end{aligned}$$

(45)

and similarly,

(46)

and hence $(P)^{\dag }=(P)$. The adjunction is an involution of $\mathrm {G}_n$. In the case of $\mathrm {G}_n$ over the real numbers the adjunction becomes transposition which is an involution corresponding to the transformation $v\rightarrow \sum v^i e^i$, $\alpha \rightarrow \sum \alpha _i e_i$. In the complex case the involution corresponds to the transformation $v\rightarrow \sum (v^i)^*e_i^{\dag }$ associated to a unitary metric $g_{ij}=h_{ij}$ [50].

The other operation is contraction. Recall that the standard operation of contraction defined for tensors fields shrinks an (r, s) tensor to an $(r-1,s-1)$ tensor. For example, for (1, 1) tensor fields on a manifold M, the contraction shrinks into functions. It is defined as the ${\mathfrak {F}}(M)$-linear map

$$\begin{aligned} {\mathbf {C}}: {\mathfrak {T}}^1_1\rightarrow {\mathfrak {F}}(M), \qquad {\mathbf {C}}(X\otimes \theta )=\langle \theta , X\rangle , \end{aligned}$$

for all one-forms $\theta$ and vector fields X. In the local chart $(U,x^1\cdots x^n)$ we must have ${\mathbf {C}}(dx^i\otimes \partial _j)=\delta ^i_j$. So for a tensor field $A \in {\mathfrak {T}}^1_1$, we obtain

$$\begin{aligned} {\mathbf {C}}\left( \sum A_{\;\;i}^j\partial _j \otimes dx^i \right) =\sum A^i_{\;\;i}=A_{\;\;i}^i. \end{aligned}$$

Einstein summation rule has been employed.

In analogy with this operation on tensors, we define a similar operation of contraction in the algebra $\mathrm {G}_n$. For example, for the elements $A_i^j ({}^{i}\!P_{\!j})$ we define:

$$\begin{aligned} {\mathbf {C}}[A_i^{\;\;j} ({}^{i}\!P_{\!j})]= & {} A_i^{\;\;j}{\mathbf {C}}[{}^{i}\!P_{\!j}]:=A_i^{\;\;j}(P_{\!j})({}^{i}\!P) \nonumber \\= & {} A_i^{\;\;j} \delta ^i_{j}(P)=A^{\;\;i}_{i}(P) \end{aligned}$$

(47)

where the property (9) and the idempotency of (P) have been used. We extend the contraction operation to the basis elements of $\mathrm {G}_n$ as follows:

$$\begin{aligned} {\mathbf {C}}_{(p)}\left[ {}^{J}\!P_{\overleftarrow{K}}\right] := \Delta ^{J}_{K}(P) \end{aligned}$$

(48)

with $|J|=|K|=p$. This full contraction shrinks the elements ${}^{J}\!P_{\overleftarrow{K}}$ to products s(P) of the projectors P in the subspace of the scalars in $\mathrm {G}_n$ with s an element of ${\mathbb {F}}$.