Note on the O(s, t) γ-matrix

Taichiro Kugo^{∗)}

§1. Euclidean Case: SO(d = 2n)

1.1. Clifford algebra C The Clifford algebra C is generated by γ_{μ} (μ = 1, ⋯, 2n):

γ_{μ}γ_{ν} + γ_{ν}γ_{μ} = 2δ_{μν},  γ_{μ}^{†} = γ_{μ}.  (1.1)

Define creation and annihilation operators of n = d/2 fermions:

\begin{aligned}a_{k}^{†} & = & \frac{1}{2}(γ_{2k−1} + iγ_{2k}) \\ a_{k} & = & \frac{1}{2}(γ_{2k−1} − iγ_{2k}) \end{aligned}  \begin{aligned}γ_{2k−1} & = & a_{k}^{†} + a_{k} \\ γ_{2k} & = & (a_{k}^{†} − a_{k})/i \end{aligned}  (1.2)

For the case of a single spece of fermion, the creation and annihilation operators a, a^{†} are represented by the Pauli matrices as follows:

{|1⟩, |2⟩} ≡ {|+⟩, |−⟩ | a^{†}|+⟩ = 0, a^{†}|−⟩ = |+⟩}
a^{†}( |+⟩, |−⟩ ) = ( |+⟩, |−⟩ ) ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) = ( |+⟩, |−⟩ )\frac{1}{2}(σ_{1} + iσ_{2})
a( |+⟩, |−⟩ ) = ( |+⟩, |−⟩ )\frac{1}{2}(σ_{1} − iσ_{2})  (1.3)

The representation space in this case is, therefore, given by:

\{|±, ±, ⋯, ±⟩\} = \{|s_{1}, s_{2}, ⋯, s_{n}⟩ = a1^{\frac{1−s_{1}}{2}} a2^{\frac{1−s_{2}}{2}} ⋯ an^{\frac{1−sn}{2}} |+, +, ⋯, +⟩\}  (1.4)

On this basis, γ matrices are represented as (Standard Representation)

γ_{1} = σ_{1} ⊗ 1 ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ 1
γ_{2} = σ_{2} ⊗ 1 ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ 1
γ_{3} = σ_{3} ⊗ σ_{1} ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ 1
γ_{4} = σ_{3} ⊗ σ_{2} ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ 1
⋮ ⋮
γ_{2n−1} = σ_{3} ⊗ σ_{3} ⊗ σ_{3} ⊗ ⋯ ⊗ σ_{3} ⊗ σ_{1}
γ_{2n} = σ_{3} ⊗ σ_{3} ⊗ σ_{3} ⊗ ⋯ ⊗ σ_{3} ⊗ σ_{2}  (1.5)

∗) kugo@gauge.scphys.kyoto-u.ac.jp

Then Γ_{5} = γ_{2n+1} is defined by

Γ_{5} = σ_{3} ⊗ σ_{3} ⊗ σ_{3} ⊗ ⋯ ⊗ σ_{3}
= i^{−n}γ_{1}γ_{2} ⋯ γ_{2n} ≡ γ_{2n+1}.  (1.6)

1.2. Charge conjugation matrix

C^{−1}γ_{μ}C = η′γ_{μ}^{T} (η′ = ±1)
C^{T} = ε′C ,  C^{†}C = 1 .  (1.7)

In even dimension, either sign for η′ can be chosen, but it is determined in the odd dimension d = 2n + 1: indeed, the relatin C^{−1}γ_{μ}C = η′γ_{μ}^{T} should hold also for μ = 2n + 1, so

C^{−1}γ_{2n+1}C = (η′)^{2n}i^{−n}γ_{1}^{T}γ_{2}^{T} ⋯ γ_{2n}^{T} = (η′)^{2n}i^{−n}(γ_{2n}γ_{2n−1} ⋯ γ_{1})^{T}
= (η′)^{2n}i^{−n}(−1)^{n(2n−1)}(γ_{1}γ_{2} ⋯ γ_{2n})^{T} = (−1)^{n}γ_{2n+1}^{T}  (1.8)

so that

η′ = (−1)^{n} = (−1)^{[ \frac{d}{2} ]} in d = 2n + 1 dimension.  (1.9)

Noting

σ_{1}σ_{i}σ_{1} = +σ_{i}^{T} and σ_{2}σ_{i}σ_{2} = −σ_{i}^{T} for i = 1, 2
σ_{1}σ_{3}σ_{1} = −σ_{3}^{T} and σ_{2}σ_{3}σ_{2} = −σ_{3}^{T} ,  (1.10)

We see that C is explicitly given by in the Standard Representation:

\{ \begin{matrix}C = σ_{1} ⊗ σ_{2} ⊗ σ_{1} ⊗ ⋯ & for η′ = +1 \\ C = σ_{2} ⊗ σ_{1} ⊗ σ_{2} ⊗ ⋯ & for η′ = −1 \end{matrix}  (1.11)

Eqs. (1.10) and (1.6) imply that the last factor of C has to be σ_{2} in the case of odd dimensions, and this requires again η′ = (−1)^{n}, i.e., Eq. (1.9). Note that this explicit C in the standard repr. is clearly unitary. The transpose is found to be:

\{ \begin{matrix}C^{T} = σ_{1} ⊗ −σ_{2} ⊗ σ_{1} ⊗ ⋯ & for η′ = +1 \\ C^{T} = −σ_{2} ⊗ σ_{1} ⊗ −σ_{2} ⊗ ⋯ & for η′ = −1 \end{matrix}  (1.12)

so that

C^{T} = C × \{ \begin{matrix}+ & for n = 1 \\ − & for n = 2 \\ − & for n = 3 \\ + & for n = 4 \end{matrix} for η′ = +1 C^{T} = C × \{ \begin{matrix}− & for n = 1 \\ − & for n = 2 \\ + & for n = 3 \\ + & for n = 4 \end{matrix} for η′ = −1

(1.13)

Thus the sign ε′ of C^{T} = ε′C is given in dimension d = 2n and 2n + 1 by

ε′ = cos \frac{π}{2} n + η′ sin \frac{π}{2} n .  (1.14)

The symmetry property of rank r gamma tensor γ_{μ_{1}μ_{2}···μ_{r}}C can be seen as follows:

γ_{μ_{1}μ_{2}···μ_{r}}C = γ_{μ_{1}}γ_{μ_{2}} ⋯ γ_{μ_{r}}C = (η′)^{r}Cγ_{μ_{1}}^{T}γ_{μ_{2}}^{T} ⋯ γ_{μ_{r}}^{T}
= (η′)^{r}ε′C^{T}γ_{μ_{1}}^{T}γ_{μ_{2}}^{T} ⋯ γ_{μ_{r}}^{T} = (η′)^{r}ε′ (γ_{μ_{r}} ⋯ γ_{μ_{2}}γ_{μ_{1}}C)^{T}
= (η′)^{r}ε′(−)^{[ \frac{r}{2} ]} (γ_{μ_{1}}γ_{μ_{2}} ⋯ γ_{μ_{r}}C)^{T} = (η′)^{r}ε′(−)^{[ \frac{r}{2} ]} (γ_{μ_{1}μ_{2}···μ_{r}}C)^{T} (1.15)

Table I. Possible signs for η′ and ε′: C^{−1}γ_{μ}C = η′γ_{μ}^{T}, C^{T} = ε′C
dimension (mod 8)  0  1  2  3  4  5  6  7
η′  + − + + − − + − + + − −
ε′  + + + + − − − − − − + +

Table II. The rank r of γ_{μ_{1}μ_{2}···μ_{r}} for which γ_{μ_{1}μ_{2}···μ_{r}}C are symmetric and anti-symmetric matrices.

dimension d η′ ε′ r of Symmetric γ_{μ_{1}μ_{2}···μ_{r}}C r of Anti-symmetric γ_{μ_{1}μ_{2}···μ_{r}}C
1  + +  0
2  + +  0, 1  2
− −  1, 2  0
3  − −  1  0
4  + −  2, 3  0, 1, 4
− −  1, 2  0, 3, 4
5  + −  2  0, 1
6  + −  2, 3, 6  0, 1, 4, 5
− +  0, 3, 4  1, 2, 5, 6
7  − +  0, 3  1, 2
8  + +  0, 1, 4, 5, 8  2, 3, 6, 7
− +  0, 3, 4, 7, 8  1, 2, 5, 6
9  + +  0, 1, 4  2, 3
10  + +  0, 1, 4, 5, 8, 9  2, 3, 6, 7, 10
− −  1, 2, 5, 6, 9, 10  0, 3, 4, 7, 8
11  − −  1, 2, 5  0, 3, 4
12  + −  2, 3, 6, 7, 10, 11  0, 1, 4, 5, 8, 9, 12
− −  1, 2, 5, 6, 9, 10  0, 3, 4, 7, 8, 11, 12

§2. Clifford 代数の表現およびη′, ε′の一意性

任意の表現のγ行列を持ってきたとき、それから fermion 演算子 a_{k}, a_{k}^{†} を式 (1.2) のように作れば、そのすべての生成演算子で消える状態 |+, +, ⋯, +⟩ を必ず作れる。これがいくつもあれば、直交化して独立にしておく。そのそれぞれの上で、式 (1.4) の部分空間を作れて、その base に関しては、元のγ行列は、標準表示の (1.5) で表現される。よって、既約表現では、状態 |+, +, ⋯, +⟩ は一意的である。この時、 base (1.4) は正規直交系であるから、あるユニタリ行列 U が存在して、

γ_{μ}^{std} = U^{−1}γ_{μ}U  (2.1)

と書ける。

この式より、また、 η′, ε′の一意性が言える。実際、一般の表示のγ行列に対する、 charge conjugation matrix C と、標準表示のそれ C^{std}との関係は、それぞれの定義を比較して

C = UC^{std}U^{T}  (2.2)

となり、 η′, ε′は、両者で共通である事がわかる。また、 C^{std} のユニタリ性から C のユニタリ性もでる。

From the transformation property C = UC^{std}U^{T} under the change of the basis, we note that C can always be taken to be 1 whenever ε′ = 1, i.e., C^{T} = C. Indeed, for such cases, C^{std} is symmetric and cntains even number of σ_{2} factors. So it is real symmetric matrix and can be be diagonalized by an orthogonal matrix O. But the eigenvalues are 1 and −1:

OC^{std}O^{T} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) = J ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) J
J ≡ ( \begin{matrix}1 & 0 \\ 0 & i \end{matrix} ) = J^{T} .  (2.3)

Therefore (JO)C^{std}(JO)^{T} becomes a unit matrix 1.

The explicit construction of such γ matrx representation for which C becomes unit matrix is as follows: in 9 dimensions, γ matrices have to be symmetric by themselves if C = 1.

γ_{1} = σ_{3} ⊗ 1 ⊗ 1 ⊗ 1
γ_{2} = σ_{1} ⊗ 1 ⊗ 1 ⊗ 1
γ_{3} = σ_{2} ⊗ σ_{3} ⊗ 1 ⊗ σ_{2}
γ_{4} = σ_{2} ⊗ σ_{1} ⊗ 1 ⊗ σ_{2}
γ_{5} = σ_{2} ⊗ σ_{2} ⊗ σ_{3} ⊗ 1
γ_{6} = σ_{2} ⊗ σ_{2} ⊗ σ_{1} ⊗ 1

γ_{7} = σ_{2} ⊗ 1 ⊗ σ_{2} ⊗ σ_{3}
γ_{8} = σ_{2} ⊗ 1 ⊗ σ_{2} ⊗ σ_{1}
γ_{9} = σ_{2} ⊗ σ_{2} ⊗ σ_{2} ⊗ σ_{2}

(2.4)

We can go down to 7 dimension, by throwing away the first two gamma matrices γ_{1}, γ_{2} and simultaneously the first column of the tensor product. Then the resultant seven γ matrices become antisymmetric by losing their first σ_{2} factors, being in accord with η′ = −1 in seven dimension. Clearly these can be repeated for 8n+1 dimensions by using these γ_{1} – γ_{8} blocks and ⊗^{4}σ_{2} and ⊗^{4}1 as a building blocks of tensor product.

§3. (General) Minkowski Case: SO(t, s)

3.1. Clifford algebra C The Clifford algebra C in this SO(t, s) case is:

γ_{μ}γ_{ν} + γ_{ν}γ_{μ} = 2η_{μν}, η_{μν} = diag(\overbrace{+, ⋯, +}^{t}, \overbrace{−, ⋯, −}^{s})

γ_{μ}^{†} = \{ \begin{matrix}γ_{μ} & for μ = 1, ⋯, t \\ −γ_{μ} & for μ = t + 1, ⋯, t + s \end{matrix} .  (3.1)

The standard representation in this case is simply given by

γ_{μ} ≡ \{ \begin{matrix}γ_{μ}^{E} & for μ = 1, ⋯, t \\ i^{−1}γ_{μ}^{E} & for μ = t + 1, ⋯, t + s \end{matrix} .  (3.2)

by putting i^{−1} to the space components from the previous Euclidean one γ_{μ}^{E}. Since the change from Euclidean to Minkowskian cases is only the multiplicatin of the non-matrix factor i, the same charge conjugation matrix C as before satisfies

C^{−1}γ_{μ}C = η′γ_{μ}^{T} (η′ = ±1).  (3.3)

So the signs of η′ and ε′ in Table I and the symmetry properties of gamma matrices in Table II are also valid in the general Minkowskian cases.

3.2. B-Conjugation

With a matrix

Γ_{0} ≡ γ_{1}γ_{2} ⋯ γ_{t}, Γ_{0}Γ_{0}^{†} = 1. (3.4)

we define the Dirac conjugate field ψ̄ by

ψ̄ = ψ^{†}Γ_{0}^{−1}  (3.5)

and Dirac conjugation by

(ψ̄γ_{μ}χ)^{†} ≡ χ̄γ̄_{μ}ψ ⇒ γ̄_{μ} = Γ_{0}γ_{μ}^{†}Γ_{0}  (3.6)

for which we have

Γ_{0}γ_{μ}^{†}Γ_{0} = (−1)^{[ \frac{t}{2} ]+t+1}γ_{μ}.  (3.7)

For the existence of Majorana(-Weyl) spinor, however, more important than the charge conjugation matrix C is the following matrix B:

B^{−1}γ_{μ}B = ηγ_{μ}^{∗} (η = ±1),
B^{T} = εB,  B^{†}B = 1.  (3.8)

Indeed, we define the charge conjugation by

ψ^{c} = Cψ̄^{T} (= CΓ_{0}^{∗}ψ^{∗})  (3.9)

and also write it into the form

ψ^{c} = Bψ^{∗} .  (3.10)

Then, comparing the two expressions, we find the relation between B and C as

B = CΓ_{0}^{∗} .  (3.11)

Indeed, then, using the unitarity of C and Γ_{0}, we have the properties Eq. (3.8) of B:

B^{†}B = Γ_{0}^{T}C^{†}CΓ_{0}^{∗} = (Γ_{0}^{†}Γ_{0})^{∗} = 1,
B^{−1}γ_{μ}B = Γ_{0}^{T}C^{−1}γ_{μ}CΓ_{0}^{∗} = η′Γ_{0}^{T}γ_{μ}^{T}Γ_{0}^{∗}
= η′(Γ_{0}^{†}γ_{μ}^{†}Γ_{0})^{∗} = η′(−1)^{[ \frac{t}{2} ]}(Γ_{0}γ_{μ}^{†}Γ_{0})^{∗} = η′(−1)^{t+1}γ_{μ}^{∗} ,
B^{T} = Γ_{0}^{†}C^{T} = ε′Γ_{0}^{†}C = ε′γ_{t} ⋯ γ_{1}C
= ε′(η′)^{t}Cγ_{t}^{T} ⋯ γ_{1}^{T} = ε′(η′)^{t}Cγ_{t}^{∗} ⋯ γ_{1}^{∗} = ε′(η′)^{t}C(γ_{t} ⋯ γ_{1})^{∗}
= ε′(η′)^{t}(−1)^{[ \frac{t}{2} ]}CΓ_{0}^{∗} = ε′(η′)^{t}(−1)^{[ \frac{t}{2} ]}B  (3.12)

so that we find

η = η′(−1)^{t+1},  ε = ε′(η′)^{t}(−1)^{[ \frac{t}{2} ]} = ε′(η)^{t}(−1)^{[ \frac{t}{2} ]} .  (3.13)

ε is a mod 4 function of t. Examining all cases by using the expression Eq. (1.14), we find

ε = cos \frac{π}{2} \frac{s − t}{2} − η sin \frac{π}{2} \frac{s − t}{2} .  (3.14)

As the previous Eq. (1.14) does, this applies only to even dimensions d = s + t = 2n for which (s − t)/2 is an integer. This is again a mod 4 function of (s − t)/2.

We thus obtain the results for the signs η and ε as follows:

ε = +1, η = −1 : s − t = 1, 2, 8, mod 8
ε = +1, η = +1 : s − t = 6, 7, 8, mod 8
ε = −1, η = −1 : s − t = 4, 5, 6, mod 8
ε = −1, η = +1 : s − t = 2, 3, 4, mod 8  (3.15)

This is summarized more explicitly in Table III.

Table III. Possible signs for η and ε: B^{−1}γ_{μ}B = ηγ_{μ}^{∗}, B^{T} = εB,
together with the signs for η′ and ε′: C^{−1}γ_{μ}C = η′γ_{μ}^{T}, C^{T} = ε′C.
dim d (mod 8)→  0  1  2  3  4  5  6  7
η′  + − + + − − + − + + − −
ε′  + + + + − − − − − − + +
t = 0 η − + − − + + − + − − + +
ε  + + + + − − − − − − + +
t = 1 η + − + + − − + − + + − −
ε  + − + + + + − + − − − −
t = 2 η − + − − + + − + − − + +
ε  − − − − + + + + + + − −
t = 3 η + − + + − − + − + + − −
ε  − + − − − − + − + + + +

Before closing this subsection, we note that the charge conjugation ψ^{c} can also be written in the form:

ψ̄^{c} = ψ^{T}Γ_{0}^{T}C^{†} · Γ_{0}^{−1} = (η′)^{t}(−)^{[ \frac{t}{2} ]}C^{−1}Γ_{0} · Γ_{0}^{−1} .  (3.16)

Namely,

ψ̄^{c} = τψ^{T}C^{−1} ,  τ ≡ (η′)^{t}(−)^{[ \frac{t}{2} ]} = η^{t}(−)^{[ \frac{t}{2} ]} .  (3.17)

We are now in a position to discuss separately when Weyl, Majorana and Majorana-Weyl spinors can exist.

3.3. Weyl spinor

When the dimension d = s + t is even, we can define Γ_{5} in this general Minkowskian case as

Γ_{5} = i^{−(d/2)}i^{s}γ_{1}γ_{2} ⋯ γ_{d} = i^{(s−t)/2}γ_{1}γ_{2} ⋯ γ_{d} ,
Γ_{5}^{2} = 1 (Γ_{5}^{†} = Γ_{5})  (3.18)

This Γ_{5} has the B conjugation property:

B^{−1}Γ_{5}B = (−)^{(s−t)/2}Γ_{5}^{∗}  (3.19)

(For odd dimension d + 1, η, and hence ε also, is fixed: the anti-hermitian γ_{d+1} should be γ_{d+1} = i^{−1}Γ_{5}, for which Eq. (3·4) holds with η = −(−)^{(s−t)/2}.)

Whenever d is even, we have Weyl spinors: we can construct the chiral projection operator

P_{±} = \frac{1}{2} (1 ± Γ_{5}) ,  (3.20)

and P_{±}ψ ≡ ψ_{±} gives the Weyl spinors.

3.4. Majorana and Pseudo-Majorana

The Dirac equation is

(iγ^{μ}∂_{μ} − m)ψ = 0  (3.21)

and its hermitian conjugation and multiplication of (−η)B gives

(iγ^{μ}∂_{μ} − (−η)m)Bψ^{∗} = 0  (3.22)

So Bψ^{∗} satisfies the same equation as ψ if (−η)m = m. If in addition ε = +1, we can equate Bψ^{∗} with ψ:

Majorana: Bψ^{∗} = ψ ,  (if ε = +1 )  (3.23)

This is possible only when ε = +1 because Eq. (3.23) implies BB^{∗} = 1. If ε = −1 and we have two spinors ψ_{i} (i = 1, 2), then we can impose instead an “SU(2) reality” condition

SU(2) Majorana: ε_{ij}Bψ^{∗j} = ψ_{i} (if ε = −1 )  (3.24)

where ε_{ij} is the SU(2) invariant anti-symmetric tensor. If we have 2N spinors, we can instead impose USp(2N) reality condition Ω_{ij}Bψ^{∗j} = ψ_{i} by replacing the SU(2) metric ε_{ij} by USp(2N) invariant (real anti-symmetric) metric Ω_{ij}. In both cases of Majorana and USp(2N) Majorana, the condition (−η)m = m means that if η = +1 we must have m = 0. So we put the term pseudo for η = +1.

Thus we can have the following four types of Majorana spinors for the combination of the signs of η and ε:

Majorana for ε = +1, η = −1 ↔ s − t = 1, 2, 8, mod 8
pseudo-Majorana for ε = +1, η = +1 ↔ s − t = 6, 7, 8, mod 8
USp(2N) Majorana for ε = −1, η = −1 ↔ s − t = 4, 5, 6, mod 8
USp(2N) pseudo-Majorana for ε = −1, η = +1 ↔ s − t = 2, 3, 4, mod 8(3.25)

3.5. Majorana-Weyl Spinors The Weyl spinors ψ_{±} ≡ P_{±}ψ satisfying

Γ_{5}ψ_{±} = ±ψ_{±}  (3.26)

always exist for even dimension d. But this is compatible with the [SU(2)] Majorana condition, Eq. (3.23) or Eq. (3.24), only if

σ ≡ (−1)^{(s−t)/2} = 1 ⟹ s − t = 0 mod 4  (3.27)

This is because we have from the B conjugation property Eq. (3.19) of Γ_{5}

B^{−1}P_{±}B = P_{±σ}^{∗}  (3.28)

with σ = (−1)^{(s−t)/2}. So, applying the chiral projection P_{±} to the Majorana spinor condition Eq. (3.23), for instance, we would get

ψ = Bψ^{∗} ⟶ P_{±}ψ = P_{±}Bψ^{∗}
ψ_{±} = BP_{±σ}^{∗}ψ^{∗} = B(ψ_{±σ})^{∗}  (3.29)

We, therefore, see that such a reality condition on Weyl spinors can be a closed condition only if σ = +1.

Noting that ε = −1 for s − t = 4 mod 8, and ε = +1 for s − t = 8 mod 8, we thus find that we can have

(pseudo-) Majorana-Weyl for s − t = 8 mod 8
USp(2N) (pseudo-) Majorana-Weyl for s − t = 4 mod 8  (3.30)