Projection operator approach to the quantization of higher spin fields

Abstract

A general method to construct free quantum fields for massive particles of arbitrary definite spin in a canonical Hamiltonian framework is presented. The main idea of the method is as follows: a multicomponent Klein–Gordon field that satisfies canonical (anti)commutation relations and serves as an auxiliary higher spin field is introduced, and the physical higher spin field is obtained by acting on the auxiliary field by a suitable differential operator. This allows the calculation of the (anti)commutation relations, the Green functions and the Feynman propagators of the higher spin fields. In addition, canonical equations of motions, which are expressed in terms of the auxiliary variables, can be obtained also in the presence of interactions, if the interaction Hamiltonian operator is known. The fields considered transform according to the (n/2,m/2)⊕(m/2,n/2) and (n/2,m/2) representations of the Lorentz group.

Appendices

Appendix A: Representations of SL(2,ℂ)

In this appendix we discuss those parts of the representation theory of SL(2,ℂ), the covering group of the Lorentz group SO(3,1), that are relevant for this paper. The main topics are the classification of finite dimensional representations, invariant bilinear forms, generalized gamma matrices, invariant complex conjugation, useful special basis vectors and their orthogonality properties, completeness relations and projectors.

1.1 A.1 Classification of the finite dimensional representations of SL(2,ℂ)

The Lie algebra of SL(2,ℂ), denoted by sl(2,ℂ), is generated by M _i , N _i , i=1,2,3, which satisfy the commutation relations

(A.1)

(A.2)

(A.3)

where ϵ _ijk is completely antisymmetric and ϵ ₁₂₃=1. We consider representations of this algebra in complex vectors spaces.

In Hilbert spaces M _i and N _i are represented by anti-Hermitian operators, which we denote by $\check{M}_{i}$, $\check{N}_{i}$. The usual Hermitian generators are $\mathrm{i}\check{M}_{i}$, $\mathrm{i}\check{N}_{i}$. The commutators of $\check{M}_{i}$, $\check{N}_{i}$ and the Hermitian generators of the translations H, P _i are

(A.4)

(A.5)

(A.6)

(A.7)

In the case of finite dimensional representations we generally do not use distinct notation for M _i , N _i and their representations.

The finite dimensional irreducible representations of sl(2,ℂ) in complex vector spaces can be labeled by two half-integers: (n/2,m/2), n,m=0,1,2,…. The (complex) dimension of the representation (n/2,m/2) is $(n+1)\*(m+1)$. The tensor product of two irreducible representations can be decomposed into a sum of irreducible representations according to the rule

(A.8)

The dual of the representation (n/2,m/2) is equivalent to the representation (n/2,m/2), the complex conjugate of (n/2,m/2) is equivalent to (m/2,n/2), and thus the adjoint of (n/2,m/2) is also equivalent to (m/2,n/2).

As is also well known, the finite dimensional irreducible representations of su(2) in complex vector spaces can be labeled by a single half-integer: (n/2), n=0,1,2,…. The (complex) dimension of the representation (n/2) is n+1. The tensor product of two irreducible representations can be decomposed into a sum of irreducible representations according to the rule

$$ \biggl(\frac{n_1}{2} \biggr)\otimes \biggl(\frac{n_2}{2} \biggr) = \bigoplus_{n_3=|n_1-n_2|}^{n_1+n_2} \biggl(\frac{n_3}{2} \biggr). $$

(A.9)

The dual, the complex conjugate and the adjoint of the representation (n/2) is equivalent to the same representation (n/2).

A representation (n/2,m/2) of sl(2,ℂ) is also a representation of the su(2) subalgebra generated by M ₁, M ₂, M ₃, and as such it is equivalent to the representation (n/2)⊗(m/2).

The left and right handed Weyl spinors are vectors (in the general linear algebraic sense) that transform according to the two-dimensional representations (1/2,0) and (0,1/2), respectively. Since these are complex representations, i.e. their complex conjugates are not equivalent to themselves, it is not possible in these representations to find a basis with respect to which the generators M _i , N _i are real matrices. The representations (n/2,m/2), n≠m are also complex in this sense, whereas (n/2,n/2) and (n/2,m/2)⊕(m/2,n/2) are real representations. (1/2,1/2) is the (complexified) Minkowski representation. To be specific, we take the following matrices in the Minkowski representation for the generators M _i , N _i :

(A.10)

(A.11)

We call the vectors transforming under the representations (n/2,0) and (0,n/2) left and right handed Weyl multispinors, respectively. (n/2,0) and (0,n/2) can be obtained as symmetric tensor products of (1/2,0) or (0,1/2), respectively:

(A.12)

(A.13)

Furthermore,

$$ (n/2,m/2)= (n/2,0)\otimes(0,m/2) . $$

(A.14)

We introduce the notations D ⁽ⁿ⁾, D ^(n,m), $\tilde{D}^{(n)}$ as

(A.15)

(A.16)

(A.17)

where n≥m. In the case n=m it should be kept in mind that D ^(n,n) is composed of two distinct copies of (n/2,n/2), although this will not be mentioned explicitly in the following. D ⁽¹⁾=(1/2,0)⊕(0,1/2) is known as the Dirac representation. The representations D ⁽ⁿ⁾, D ^(n,m), $\tilde{D}^{(n)}$ are real, i.e. it is possible to find basis vectors such that M _i , N _i are real matrices.

1.2 A.2 Basis vectors for the representations (n/2,m/2)

A specific matrix representation of M _i , N _i on left and right handed Weyl spinors can be given as follows. For (1/2,0),

$$ M_{i}=\frac{-\mathrm{i}}{2}\sigma_{i} ,\qquad N_{i}=\frac{1}{2}\sigma_{i} ,\quad i=1,2,3, $$

(A.18)

for (0,1/2),

$$ M_{i}=\frac{-\mathrm{i}}{2}\sigma_{i} ,\qquad N_{i}=-\frac{1}{2}\sigma_{i} ,\quad i=1,2,3 , $$

(A.19)

where σ _i are the Pauli sigma matrices

$$ \everymath{\displaystyle} \begin{array}{@{}l} \sigma_1= \left ( \begin{array}{c@{\quad}c} 0 & 1\\ 1 & 0\\ \end{array} \right ), \\\noalign{\vspace{6pt}} \sigma_2= \left ( \begin{array}{c@{\quad}c} 0 & -\mathrm{i}\\ \mathrm{i}& 0\\ \end{array} \right ), \\\noalign{\vspace{6pt}} \sigma_3= \left ( \begin{array}{c@{\quad}c} 1 & 0\\ 0 & -1\\ \end{array} \right ), \end{array} $$

(A.20)

which have the well known properties

$$ [\sigma_{i},\sigma_{j}]=2\mathrm{i}\epsilon_{ijk} \sigma_{k} ,\qquad \sigma_{i}\sigma_{j}= \mathrm{i}\epsilon_{ijk}\sigma_{k}, \qquad \operatorname{Tr}\sigma_{i}=0. $$

(A.21)

In the following e ₁, e ₂ and e ₃, e ₄ denote fixed basis vectors of (1/2,0) and (0,1/2) with respect to which M _i , N _i take the form above (e ₁, e ₂ are the basis vectors of (1/2,0) and e ₃, e ₄ are the basis vectors of (0,1/2)). We call e ₁, e ₂, e ₃, e ₄ Weyl basis vectors. The corresponding dual basis vectors are denoted by $\hat{e}_{i}$, i=1,…,4. These vectors are defined by the property $(\hat{e}_{i})^{\alpha}(e_{j})_{\alpha}= \delta_{ij}$, i,j=1,2, $(\hat{e}_{i})^{\dot{\alpha}} (e_{j})_{\dot{\alpha}} = \delta_{ij}$, i,j=3,4. Here and throughout the paper dotted indices are used for right handed spinors.

Basis vectors for (n/2,0) and (0,n/2) can be obtained by taking symmetric tensor products of e ₁, e ₂ and e ₃, e ₄. Specifically, we define the basis vectors

(A.22)

(A.23)

Here we have introduced the notation Sym for symmetrization, which means summation over all permutations of the indices in the subscript. In the formulas above this means that there are n! terms on the right hand side, of which many are actually identical. The dual basis vectors $\hat{E}_{l}$, $\hat{F}_{l}$ are obtained by replacing the vectors e _i by $\hat{e}_{i}$ in these formulas.

The vectors E _l ⊗F _k , l=1,…,(n+1), k=1,…,(m+1) constitute a basis for (n/2,m/2), and the vectors $\hat{E}_{l}\otimes\hat{F}_{k}$ are dual basis vectors.

The action of M _i , N _i on the elements of (n/2,0) and (0,n/2) is defined in the usual way, i.e. if

$$v=\sum_{i_1,i_2,\dots, i_n} c_{i_1,i_2,\dots, i_n} e_{i_1}\otimes e_{i_2}\otimes\dots\otimes e_{i_n}, $$

where $c_{i_{1},i_{2},\dots, i_{n}}$ are complex coefficients, then

(A.24)

and the same formula applies also to N _i . The definition is similar for (n/2,m/2); if v=∑ _l,k c _l,k E _l ⊗F _k , then M _i v=∑ _l,k c _l,k [(M _i E _l )⊗F _k +E _l ⊗(M _i F _k )].

1.3 A.3 Invariant bilinear forms

In this subsection invariant bilinear forms on the representations (n/2,m/2), D ⁽ⁿ⁾, D ^(n,m) and $\tilde{D}^{(n)}$ are introduced.

(1/2,0) and (0,1/2) admit unique (up to multiplication by constant) invariant bilinear forms, which are nondegenerate and antisymmetric. Specifically, we use the invariant bilinear forms given by the following matrices with respect to the basis vectors e ₁, e ₂ and e ₃, e ₄:

$$ \epsilon^L =\left ( \begin{array}{c@{\quad}c} 0 & -1\\ 1 & 0\\ \end{array} \right ), \qquad \epsilon^R= \left ( \begin{array}{c@{\quad}c} 0 & 1\\ -1 & 0\\ \end{array} \right ) . $$

(A.25)

(This means ϵ ^L(e ₁,e ₂)=−ϵ ^R(e ₃,e ₄)=−1.) The matrices of the inverses of ϵ ^L and ϵ ^R (with respect to the dual basis vectors) are the negative of the matrices of ϵ ^L and ϵ ^R.

(n/2,m/2) also admit unique (up to multiplication by constant) invariant bilinear forms, which are nondegenerate. Specifically, on (n/2,0) and on (0,n/2) we take the bilinear forms ϵ ^L and ϵ ^R defined as

(A.26)

(A.27)

$\epsilon^{L\alpha_{1}\alpha_{2}\dots\alpha_{n};\delta_{1}\delta_{2}\dots \delta_{n}}$ and $\epsilon^{R\dot{\alpha}_{1}\dot{\alpha}_{2}\dots\dot{\alpha }_{n};\dot{\delta}_{1}\dot{\delta}_{2}\dots\dot{\delta}_{n}}$ are completely symmetric in their indices α ₁ α ₂…α _n , δ ₁ δ ₂…δ _n , $\dot{\alpha}_{1}\dot{\alpha}_{2}\dots\dot{\alpha}_{n}$ and $\dot{\delta}_{1}\dot{\delta}_{2}\dots\dot{\delta}_{n}$. $\epsilon^{L\alpha_{1}\alpha_{2}\dots\alpha_{n};\delta_{1}\delta_{2}\dots \delta_{n}}$ and $\epsilon^{R\dot{\alpha}_{1}\dot{\alpha}_{2}\dots\dot{\alpha }_{n};\dot{\delta}_{1}\dot{\delta}_{2}\dots\dot{\delta}_{n}}$, as defined in (A.26) and (A.27), have several (1/2,0)- or (0,1/2)-indices, corresponding to the fact that (n/2,0) and (0,n/2) were defined as spaces of symmetric tensors having n (1/2,0)- or (0,1/2)-indices. Nevertheless, ϵ ^L and ϵ ^R can also be regarded as tensors having two upper (n/2,0)- or (0,n/2)-indices. In the following and in other sections of the paper we often use a notation that corresponds to this view of ϵ ^L and ϵ ^R, as we also mentioned at the end of the Introduction.

On (n/2,m/2) we take the invariant bilinear form

(A.28)

These bilinear forms are antisymmetric for n+m odd and symmetric for n+m even.

The inverses of ϵ ^L and ϵ ^R can also be obtained as

(A.29)

(A.30)

The inverse of ϵ (defined in (A.28)) is

(A.31)

The nonzero matrix elements of ϵ ^L and ϵ ^R with respect to the basis vectors E _l , F _l defined above are

(A.32)

(A.33)

The inverses of ϵ ^L and ϵ ^R have the nonzero matrix elements

(A.34)

(A.35)

with respect to the dual basis. We do not introduce distinct notation for the inverses of ϵ ^L, and ϵ ^R, but this should not cause confusion.

On D ⁽¹⁾, we take the antisymmetric invariant bilinear form ϵ that coincides with ϵ ^L and ϵ ^R on the subspaces (1/2,0) and (0,1/2), respectively, and is diagonal with respect to the decomposition D ⁽¹⁾=(1/2,0)⊕(0,1/2), i.e. that has the matrix

$$ \epsilon=\left ( \begin{array}{c@{\quad}c} \epsilon^L & 0 \\ 0 & \epsilon^R \end{array} \right ) $$

(A.36)

with respect to the basis e ₁, e ₂, e ₃, e ₄. We also take invariant bilinear forms ϵ on D ⁽ⁿ⁾ and D ^(n,m) defined in the same way.

1.4 A.4 Generalized gamma matrices

With respect to the Weyl basis, the Dirac gamma matrices are given by

$$ \gamma^0=\left ( \begin{array}{c@{\quad}c} 0 & I \\ I & 0 \end{array} \right ), \qquad \gamma^i=\left ( \begin{array}{c@{\quad}c} 0 & \sigma_i \\ -\sigma_i & 0 \end{array} \right ), \quad i=1,2,3, $$

(A.37)

where I denotes the 2×2 identity matrix. These matrices satisfy the well known identity

$$ \bigl\{\gamma^\mu,\gamma^\nu\bigr\}=2g^{\mu\nu} . $$

(A.38)

In addition, they are related to the representation M _Di , N _Di of the generators M _i , N _i on D ⁽¹⁾ by the equations

(A.39)

(A.40)

One can also define the matrix γ ⁵ as

$$ \gamma^5=\gamma^0\gamma^1 \gamma^2\gamma^3. $$

(A.41)

It satisfies the identity

$$ \bigl\{ \gamma^5,\gamma^\mu\bigr\}=0, $$

(A.42)

and its matrix form is

$$ \gamma^5=\left ( \begin{array}{c@{\quad}c} \mathrm{i}I & 0 \\ 0 & -\mathrm{i}I \\ \end{array} \right ). $$

(A.43)

γ ^μ, regarded as a tensor with an upper Minkowski vector index and an upper and a lower Dirac spinor index (the latter two are suppressed), is an SL(2,ℂ)-invariant tensor. As usual, this means

$$ {(\varLambda_M)^\nu}_\mu{( \varLambda_D)_\delta}^\alpha{\bigl( \varLambda_D^{-1}\bigr)_\beta}^\rho{\bigl( \gamma^\mu\bigr)_\alpha}^\beta= {\bigl( \gamma^\nu\bigr)_\delta}^\rho, $$

(A.44)

where Λ _M and Λ _D denote the representation of an element Λ of SL(2,ℂ) in the Minkowski and Dirac spinor spaces, respectively. γ ⁵ is also an invariant tensor having one upper and one lower Dirac spinor index.

The Dirac gamma matrices take the form

$$ \gamma^\mu=\left ( \begin{array}{c@{\quad}c} 0 & \eta^\mu\\ \bar{\eta}^\mu& 0 \\ \end{array} \right ) $$

(A.45)

with respect to the decomposition D ⁽¹⁾=(1/2,0)⊕(0,1/2), where η ^μ is an invariant tensor with one Minkowski vector index, one upper (0,1/2)-index and one lower (1/2,0)-index, and $\bar{\eta}^{\mu}$ is an invariant tensor with one Minkowski vector index, one lower (0,1/2)-index and one upper (1/2,0)-index (however, the spinor indices are suppressed here). It follows from (A.38) that η ^μ and $\bar{\eta}^{\mu}$ satisfy the identities

(A.46)

(A.47)

As generalizations of η ^μ and $\bar{\eta}^{\mu}$, the invariant tensors $\tau^{\mu_{1}\mu_{2}\dots\mu_{n}}$ and $\bar{\tau}^{\mu_{1}\mu_{2}\dots\mu_{n}}$ can be defined as

(A.48)

(A.49)

${(\tau^{\mu_{1}\mu_{2}\dots\mu_{n}})_{\alpha_{1}\alpha_{2}\dots\alpha_{n}}}^{\dot{\delta}_{1}\dot{\delta}_{2}\dots\dot{\delta}_{n}}$ and ${(\bar{\tau}^{\mu_{1}\mu_{2}\dots\mu_{n}})_{\dot{\alpha}_{1}\dot {\alpha}_{2}\dots\dot{\alpha}_{n}}}^{\delta_{1}\delta_{2}\dots\delta_{n}}$ are completely symmetric in the indices α ₁ α ₂…α _n , δ ₁ δ ₂…δ _n , $\dot{\alpha}_{1}\dot{\alpha}_{2}\dots\dot{\alpha}_{n}$ and $\dot {\delta}_{1}\dot{\delta}_{2}\dots\dot{\delta}_{n}$, therefore τ can be regarded as a tensor having n upper Minkowski indices, one upper (0,n/2)-index and one lower (n/2,0)-index, and $\bar{\tau}$ can be regarded as a tensor having n upper Minkowski indices, one lower (0,n/2)-index and one upper (n/2,0)-index. In the following and in other sections of the paper we often use a notation which corresponds to this interpretation of τ and $\bar{\tau}$. Both τ and $\bar{\tau}$ are also completely symmetric and traceless in their Minkowski indices;

$$ g_{\mu\nu}\tau^{\mu\nu\dots\lambda} = g_{\mu\nu}\bar{\tau }^{\mu\nu\dots\lambda} = 0. $$

(A.50)

The Fierz identities

(A.51)

(A.52)

can be generalized as

(A.53)

(A.54)

We also have

$$ {\bigl(\tau^{\mu_1\mu_2\dots\mu_n}\bigr)_\alpha}^{\dot{\beta}} {(\bar { \tau}_{\mu_1\mu_2\dots\mu_n})_{\dot{\gamma}}}^\rho= \frac {2^n}{(n!)^2}{ \delta_\alpha}^\rho{\delta_{\dot{\gamma}}}^{\dot {\beta}} . $$

(A.55)

For (n/2,m/2), n≠0, m≠0, n≥m, one can define the invariant tensor

(A.56)

which can be regarded as a tensor with n+m upper Minkowski indices, one upper (m/2,n/2)-index and one lower (n/2,m/2)-index. τ is obviously completely symmetric and traceless in the first n and last m Minkowski indices. We introduce the notation $\bar{\tau}$ for the tensor defined in the same way as τ, but with n≠0, m≠0, n≤m.

Generalized gamma tensors for D ^(n,m) can be defined as

(A.57)

where the matrix form corresponds to the decomposition D ^(n,m)=(n/2,m/2)⊕(m/2,n/2). $\gamma^{\mu_{1}\mu_{2}\dots\mu_{n}\mu'_{1}\mu'_{2}\dots\mu'_{m}}$ is an invariant tensor having n+m upper Minkowski indices, one upper D ^(n,m)-index and one lower D ^(n,m)-index. Generalized gamma tensors for D ⁽ⁿ⁾ can be defined in the same way (with m=0).

We also introduce the tensors $\gamma_{(1)}^{\mu_{1}\mu_{2}\dots\mu_{n-1}\mu'_{1}\mu'_{2}\dots\mu'_{m-1}},\allowbreak \gamma_{(2)}^{\mu_{1}\mu_{2}\dots\mu_{n-2}\mu'_{1}\mu'_{2}\dots\mu'_{m-2}}, \dots, \gamma_{(m)}^{\mu_{1}\mu_{2}\dots\mu_{n-m}}$ (assuming m≥1). $\gamma_{(1)}^{\mu_{1}\mu_{2}\dots\mu_{n-1}\mu'_{1}\mu'_{2}\dots\mu'_{m-1}}$ is obtained by contracting the μ ₁ and $\mu'_{1}$ Minkowski indices of $\gamma^{\mu_{1}\mu_{2}\dots\mu_{n}\mu'_{1}\mu'_{2}\dots\mu'_{m}}$ with $g_{\mu_{1}\mu_{1}'}$; $\gamma_{(2)}^{\mu_{1}\mu_{2}\dots\mu_{n-2}\mu'_{1}\mu'_{2}\dots\mu'_{m-2}}$ is obtained by contracting the μ ₁ and $\mu'_{1}$ Minkowski indices of $\gamma_{(1)}^{\mu_{1}\mu_{2}\dots\mu_{n-1}\mu'_{1}\mu'_{2}\dots\mu'_{m-1}}$ with $g_{\mu_{1}\mu_{1}'}$, and so on. The tensors

and

can be defined similarly. Any further contractions of $\tau_{(m)}^{\mu_{1}\mu_{2}\dots\mu _{n-m}}$ and $\bar{\tau}_{(m)}^{\mu_{1}\mu_{2}\dots\mu_{n-m}}$ (or of $\tau^{\mu_{1}\mu_{2}\dots\mu_{n}}$ and $\bar{\tau}^{\mu _{1}\mu_{2}\dots\mu_{n}}$, if m=0) with g _μν give zero.

The $\gamma^{\mu_{1}\mu_{2}\dots\mu_{n}\mu'_{1}\mu'_{2}\dots\mu'_{m}}$ tensors have the following important properties:

(A.58)

(A.59)

i.e.

$${(\gamma^{\mu_1\mu_2\dots\mu_n\mu'_1\mu'_2\dots\mu '_m})_\alpha}^\rho\epsilon_{\rho\beta} $$

and

$${(\gamma^{\mu_1\mu_2\dots\mu_n\mu'_1\mu'_2\dots\mu'_m})_\rho }^\alpha\epsilon^{\rho\beta} $$

are symmetric in α and β.

also obviously have this property.

A further important property of the γ tensors is

(A.60)

for any vector k _μ . The related identity

(A.61)

is of central importance in Sect. 5.

We define γ ⁵ on D ⁽ⁿ⁾ and on D ^(n,m) as

$$ \gamma^5=\left ( \begin{array}{c@{\quad}c} \mathrm{i}I & 0 \\ 0 & -\mathrm{i}I \\ \end{array} \right ), $$

(A.62)

where the matrix form is understood with respect to the decomposition D ⁽ⁿ⁾=(n/2,0)⊕(0,n/2) and D ^(n,m)=(n/2,m/2)⊕(m/2,n/2). γ ⁵ anticommutes with $\gamma^{\mu_{1}\mu_{2}\dots\mu_{n}}$ and $\gamma^{\mu_{1}\mu_{2}\dots\mu_{n}\mu'_{1}\mu'_{2}\dots\mu'_{m}}$, $\gamma_{(1)}^{\mu_{1}\mu_{2}\dots\mu_{n-1}\mu'_{1}\mu'_{2}\dots\mu'_{m-1}}, \dots, \gamma_{(m)}^{\mu_{1}\mu_{2}\dots\mu_{n-m}}$. γ ⁵ is not defined for $\tilde{D}^{(n)}$.

1.5 A.5 Invariant complex conjugation, basis vectors and projectors in D ⁽ⁿ⁾, D ^(n,m) and $\tilde{D}^{(n)}$

1.5.1 A.5.1 The representations D ⁽ⁿ⁾

In the following we focus on the representations (n/2,0), (0,n/2) and D ⁽ⁿ⁾, and we return to D ^(n,m) and $\tilde{D}^{(n)}$ subsequently in Appendices A.5.2 and A.5.3.

The action of γ ^00…0 as a linear mapping on D ⁽ⁿ⁾ is given by

(A.63)

(A.64)

We also have γ ⁵ E _l =iE _l , γ ⁵ F _l =−iF _l , l=1,…,(1+n).

A complex conjugation can be defined on D ⁽¹⁾ in the following way:

$$ e_1^*= -e_4,\qquad e_2^*= e_3, $$

(A.65)

and the complex conjugate of an arbitrary Dirac spinor c _i e _i , c _i ∈ℂ, is given by $(c_{i}e_{i})^{*}=c_{i}^{*}e_{i}^{*}$. This is an invariant complex conjugation in the sense that it commutes with the action of M _i , N _i . The complex conjugate of the dual basis vectors is defined in the same way:

$$ \hat{e}_1^*= -\hat{e}_4, \qquad\hat{e}_2^*=\hat{e}_3. $$

(A.66)

Real basis vectors (with respect to the complex conjugation above) can be defined as follows:

$$ \everymath{\displaystyle} \begin{array}{@{}l} v_1=\frac{e_2+e_3}{\sqrt{2}},\qquad v_2=\frac{\mathrm{i}e_1+\mathrm{i}e_4}{\sqrt{2}}, \\\noalign{\vspace{7pt}} v_3=\frac{-e_1+e_4}{\sqrt{2}},\qquad v_4=\frac{\mathrm{i}e_2-\mathrm{i}e_3}{\sqrt{2}}. \end{array} $$

(A.67)

The representations of M _i , N _i on D ⁽¹⁾ are real matrices with respect to this basis, and the gamma matrices are imaginary:

γ ⁵ takes the form

$$\gamma^5= \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ \end{array} \right ) . $$

ϵ has the canonical form

$$\epsilon= \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & \mathrm{i}& 0 & 0 \\ -\mathrm{i}& 0 & 0 & 0\\ 0 & 0 & 0 & \mathrm{i}\\ 0 & 0 & -\mathrm{i}& 0\\ \end{array} \right ) . $$

The complex conjugation defined on D ⁽¹⁾ gives rise to the following SL(2,ℂ)-invariant complex conjugation on D ⁽ⁿ⁾:

(A.68)

(A.69)

This can be further written as

(A.70)

(A.71)

The complex conjugates of the dual basis vectors $\hat{E}_{l}$, $\hat {F}_{l}$ are given by the same formulas, i.e.

(A.72)

(A.73)

On real vectors, ϵ is purely imaginary if n is odd and real if n is even. As mentioned after (A.27), ϵ is also symmetric if n is even, therefore it has a well-defined signature, which is (n+1,n+1). The tensors ${(\gamma^{\mu_{1}\mu_{2}\dots\mu_{n}})_{\alpha}}^{\beta}$ are also imaginary if n is odd and real if n is even. γ ⁵ is real for any value of n.

The following relations are also important:

(A.74)

(A.75)

(A.76)

(A.77)

(A.78)

(A.79)

(A.80)

(A.81)

(A.82)

(A.83)

We introduce the basis vectors

(A.84)

(A.85)

and their duals

(A.86)

(A.87)

We also define the vectors u _i (k), v _i (k) as

(A.88)

(A.89)

where $\varLambda_{D^{(n)}}(k)$ represents in D ⁽ⁿ⁾ the unique SL(2,ℂ) element Λ(k) determined by the properties that Λ(k) is a continuous function of k, Λ(0)=I, and the Lorentz transformation corresponding to Λ(k) is the Lorentz boost that takes the dual four-vector (μ,0) to (ω(k),k). The vectors dual to u _i (k), v _i (k) are denoted by $\hat{u}_{i}(k)$, $\hat{v}_{i}(k)$.

The vectors $\hat{u}_{i}(k)$ and $\hat{v}_{i}(k)$ satisfy the orthogonality relations

$$ \everymath{\displaystyle} \begin{array}{@{}l} \bigl\langle\hat{u}_i(k) , \hat{u}_j(k) \bigr\rangle= \delta_{ij}, \\\noalign{\vspace{7pt}} \bigl\langle\hat{v}_i(k) ,\hat{v}_j(k) \bigr\rangle =-\delta_{ij}, \\\noalign{\vspace{7pt}} \bigl\langle\hat{u}_i(k) , \hat{v}_j(k) \bigr\rangle=0, \end{array} $$

(A.90)

where 〈 , 〉 denotes the scalar product introduced at the beginning of Sect. 2.1.

The following obvious completeness relations are also important to note:

(A.91)

(A.92)

The second relation is obtained from the first one by complex conjugation. The SL(2,ℂ)-invariant complex conjugation defined above is understood to be applied, which coincides with the componentwise complex conjugation if the indices α and β correspond to a real basis. Not only here, but also throughout the appendix we use SL(2,ℂ)-invariant complex conjugation.

The complex conjugates of u _i (k) and v _i (k) are

(A.93)

(A.94)

if n is even, and

(A.95)

(A.96)

if n is odd. The same formulas apply to the dual vectors $\hat {u}_{i}(k)$ and $\hat{v}_{i}(k)$.

The matrices

$$\frac{1}{2\mu^n} {\bigl(\mu^n+(-1)^n k_{\mu_1} k_{\mu_2}\dots k_{\mu_n}\gamma^{\mu_1\mu_2\dots\mu_n} \bigr)_\alpha}^\beta $$

and

$$\frac{1}{2\mu^n} {\bigl(\mu^n - (-1)^n k_{\mu_1} k_{\mu_2}\dots k_{\mu_n}\gamma^{\mu_1\mu_2\dots\mu_n} \bigr)_\alpha}^\beta $$

are projectors if k _μ k ^μ=μ ², specifically we have the identities

(A.97)

(A.98)

(A.99)

(A.100)

for all i=1,…,(n+1), which imply

(A.101)

(A.102)

Complex conjugation of these formulas gives

(A.103)

(A.104)

1.5.2 A.5.2 The representations D ^(n,m)

We consider now the representations D ^(n,m) with n,m≠0, n≥m. The vectors

(A.105)

are basis vectors in D ^(n,m); E _ij span the subspace (n/2,m/2), whereas F _ji span the subspace (m/2,n/2). The corresponding dual basis vectors are

(A.106)

The action of γ ^00…0 on D ^(n,m) is

(A.107)

(A.108)

γ ^00…0 commutes with $\gamma_{(1)}^{00\dots0}, \gamma_{(2)}^{00\dots0}, \ldots, \gamma_{(m)}^{00\dots0}$. The action of γ ⁵ on E _ij and F _ji is γ ⁵ E _ij =iE _ij , γ ⁵ F _ji =−iF _ji .

The complex conjugates of the basis vectors are

(A.109)

(A.110)

and the same formulas apply to the dual basis vectors. ϵ is purely imaginary on real vectors if n+m is odd and real if n+m is even. The signature of ϵ is ((n+1)(m+1),(n+1)(m+1)) when n+m is even. The τ and γ tensors are also purely imaginary if n+m is odd and real if n+m is even. γ ⁵ is real for all values of n+m.

The nonzero matrix elements of ϵ with respect to the basis vectors E _ij are

(A.111)

(A.112)

The nonzero matrix elements of the inverse of ϵ with respect to the dual basis vectors $\hat{E}_{ij}$ are

(A.113)

(A.114)

The following relations are also important to note:

(A.115)

(A.116)

(A.117)

(A.118)

(A.119)

(A.120)

(A.121)

(A.122)

(A.123)

(A.124)

We introduce the basis vectors

(A.125)

(A.126)

and their duals

(A.127)

(A.128)

We also define the vectors u _ij (k), v _ij (k) in the same way as u _i (k), v _i (k) in (A.88) and (A.89). The dual vectors $\hat{u}_{ij}(k)$ and $\hat{v}_{ij}(k)$ satisfy the orthogonality relations

$$ \everymath{\displaystyle} \begin{array}{@{}l} \bigl\langle\hat{u}_{ij}(k) , \hat{u}_{kl}(k) \bigr\rangle=\delta_{ik}\delta_{jl}, \\\noalign{\vspace{7pt}} \bigl\langle\hat{v}_{ij}(k) , \hat{v}_{kl}(k) \bigr\rangle =-\delta_{ik}\delta_{jl}, \\\noalign{\vspace{7pt}} \bigl\langle\hat{u}_{ij}(k) , \hat{v}_{kl}(k) \bigr\rangle=0, \end{array} $$

(A.129)

where 〈 , 〉 denotes the scalar product introduced at the beginning of Sect. 2.1.

The completeness relations analogous to (A.91) and (A.92) take the form

(A.130)

(A.131)

The complex conjugates of u _ij (k), v _ij (k), i=1,…,(n+1), j=1,…,(m+1), are

(A.132)

(A.133)

if n+m is even, and

(A.134)

(A.135)

if n+m is odd. The same formulas apply to the dual vectors $\hat {u}_{ij}(k)$ and $\hat{v}_{ij}(k)$.

The matrices

$$\frac{1}{2\mu^{n+m}} {\bigl(\mu^{n+m}+(-1)^{n+m} k_{\mu_1} k_{\mu _2}\dots k_{\mu_{n+m}}\gamma^{\mu_1\mu_2\dots\mu_{n+m}} \bigr)_\alpha}^\beta $$

and

$$\frac{1}{2\mu^{n+m}} {\bigl(\mu^{n+m} - (-1)^{n+m} k_{\mu_1} k_{\mu_2}\dots k_{\mu_{n+m}}\gamma^{\mu_1\mu_2\dots\mu_{n+m}}\bigr)_\alpha}^\beta $$

are projectors if k _μ k ^μ=μ ², and identities analogous to (A.97)–(A.104) hold.

The space spanned by u _ij ≡u _ij (k=0) can be decomposed into irreducible representations with respect to the SU(2) (rotation) little group generated by M ₁, M ₂, M ₃. The decomposition is ((n+m)/2)⊕((n+m)/2−1)⊕…⊕((n−m)/2). These invariant subspaces are orthogonal with respect to the scalar product 〈 , 〉 introduced in Sect. 2.1. One can also introduce orthonormal (with respect to 〈 , 〉) basis vectors in these subspaces, dual basis vectors, and then the boosted versions of these. We denote these vectors by u _(s),i (k) and $\hat{u}_{(s),i}(k)$, where s denotes the spin and i is an index labeling the basis vectors within the subspace of spin s. Projection operators on the invariant subspaces and their boosted versions can also be formed using these basis vectors and the dual basis vectors. These projection operators are $\sum_{i=1}^{2s+1} u_{(s),i\alpha}(k)\hat{u}_{(s),i}^{\beta}(k)$. Similar statements can be made also for the space spanned by v _ij .

1.5.3 A.5.3 The representations $\tilde{D}^{(n)}$

We consider finally the representations $\tilde{D}^{(n)}$. The tensors $\tau^{\mu_{1}\mu_{2}\dots\mu_{n}\mu_{1}'\mu_{2}'\dots\mu _{n}'}$ have the properties

(A.136)

(A.137)

which are analogous to (A.58) and (A.59). The formulas

(A.138)

(A.139)

which are analogous to (A.60) and (A.61), also hold. The vectors

$$ E_{ij}=E_i\otimes F_j,\quad i=1,\dots, (1+n),\ j=1,\dots, (1+n), $$

(A.140)

are basis vectors in $\tilde{D}^{(n)}$. The corresponding dual basis vectors are

$$ \hat{E}_{ij}=\hat{E}_i\otimes\hat{F}_j, \quad i=1,\dots, (1+n),\ j=1,\dots, (1+n). $$

(A.141)

The action of τ ^00…0 on $\tilde{D}^{(n)}$ is given by

$$ \tau^{00\dots0} E_{ij} = E_{ji}. $$

(A.142)

The complex conjugates of the basis vectors are defined as

(A.143)

and the same formulas apply to the dual basis vectors. ϵ is real on real vectors and is symmetric. The signature of ϵ is $(\frac{(n+2)(n+1)}{2},\frac{n(n+1)}{2} )$. The τ tensor is real.

The nonzero matrix elements of ϵ with respect to the basis vectors E _ij are

$$ \epsilon(E_{ij},E_{n-i+2,n-j+2}) = (-1)^{n+i+j}. $$

(A.144)

The nonzero matrix elements of the inverse of ϵ with respect to the dual basis vectors $\hat{E}_{ij}$ are given by the same formula,

$$ \epsilon(\hat{E}_{ij},\hat{E}_{n-i+2,n-j+2}) = (-1)^{n+i+j}. $$

(A.145)

The following relations are also worth noting:

(A.146)

(A.147)

(A.148)

(A.149)

(A.150)

The same formulas apply to the dual vectors and the inverse of ϵ.

We introduce the basis vectors

(A.151)

(A.152)

(A.153)

and their duals

(A.154)

(A.155)

(A.156)

We also define the vectors u _ij (k), v _ij (k) in the same way as u _i (k), v _i (k) in (A.88) and (A.89).

The complex conjugates of u _ij (k), v _ij (k) are

(A.157)

(A.158)

The same formulas apply to the dual vectors $\hat{u}_{ij}(k)$ and $\hat{v}_{ij}(k)$.

The dual vectors $\hat{u}_{ij}(k)$ and $\hat{v}_{ij}(k)$ satisfy the orthogonality relations

$$ \everymath{\displaystyle} \begin{array}{@{}l} \bigl\langle\hat{u}_{ij}(k) , \hat{u}_{kl}(k) \bigr\rangle=\delta_{ik}\delta_{jl}, \\\noalign{\vspace{7pt}} \bigl\langle \hat{v}_{ij}(k) , \hat{v}_{kl}(k) \bigr\rangle =-\delta_{ik}\delta_{jl}, \\\noalign{\vspace{7pt}} \bigl\langle \hat{u}_{ij}(k) , \hat{v}_{kl}(k) \bigr\rangle=0, \end{array} $$

(A.159)

where 〈 , 〉 denotes the scalar product introduced at the beginning of Sect. 2.1.

The space spanned by u _ij ≡u _ij (k=0) can be decomposed into irreducible representations with respect to the SU(2) little group generated by M ₁, M ₂, M ₃. The decomposition is (n)⊕(n−2)⊕(n−4)⊕…. These invariant subspaces are orthogonal with respect to the scalar product 〈 , 〉. One can also introduce orthonormal (with respect to 〈 , 〉) basis vectors in these subspaces, dual basis vectors, and the boosted versions of these. Projection operators on the invariant subspaces and their boosted versions can also be formed using these basis vectors and the dual basis vectors. The decomposition of the space spanned by v _ij is (n−1)⊕(n−3)⊕(n−5)⊕…; otherwise it can be treated in the same way as the space spanned by u _ij .

Appendix B: Equal-time anticommutator of the spin-3/2 field transforming according to (3/2,0)⊕(0,3/2)

In this appendix the equal-time anticommutator $[\psi_{\alpha}(x,t),\allowbreak \psi_{\beta}^{\dagger}(y,t)]_{+}$ is calculated in the case when ψ transforms according to the representation (3/2,0)⊕(0,3/2), with the aim of illustrating the general rule described in Sect. 5.1. The anticommutator of the Dirac field is calculated in the same way in Sect. 6.

According to (175) we have

(B.1)

By applying (178) and considering the equal-time anticommutation relations of Ψ, only those terms on the right hand side give nonzero contribution which contain an odd number of time derivatives, thus

(B.2)

Taking into account the anticommutation relations (38) and (39) of Ψ, we get the final result

(B.3)

In this formula the derivations on the right hand side are understood to be derivations with respect to the components of x.