88
5. PARTICLE
which has γ-matrix type commutation relations. It anticommutes with
∂
ˆ
γ = γ − ζ =
− 1ζ ,
(5.3.7b)
∂ζ
2
which is an independent set of γ-matrices. However, it is γ which appears in the
Dirac equation, obtained by varying ψ.
In a light-cone formalism, we again eliminate all auxiliary “−” components by
their equations of motion, and use the gauge invariance (5.3.2-3) to fix the “+”
components
X+ = p+τ
→
x+ = p+τ
,
γ+ = 0 .
(5.3.8)
We then find G = 1, and (5.3.6) reduces to
L = .x
2
.
.
−p+ + .
xipi − 1p
ζ
iζ ζ
2 i − i(γ− − ζ −).
ζ+ − iγi i + 12 i i .
(5.3.9)
In order to obtain the usual spinor field, it’s necessary to add a lagrange multiplier term to the action constraining ˆ
γ = 0. This constraint can either be solved classically
(but only for even spacetime dimension D) by determining half of the ζ’s to be
the canonical conjugates of the other half (consider ζ1 + iζ2 vs. ζ1 − iζ2, etc.), or by imposing it quantum mechanically on the field Gupta-Bleuler style. The former
approach sacrifices manifest Lorentz invariance in the coordinate approach; however, if the γ’s are considered simply as operators (without reference to their coordinate representation), then the field is the usual spinor representation, and both can be represented in the usual matrix notation. This constrained action is equivalent to the second-order action
S =
dτ dθ 1G−1(
2
D2X) · (DX)
=
dτ dθ 1(GdX)d(GdX) ,
2
=
dτ ( 1g−1.
x2 + ig−1ψγ
iγ
2
· .x − 12
· .γ) ,
(5.3.10a)
or, in first-order form for x only,
S =
dτ (.x · p − g 1p2 + 1i.γ
2
2
· γ + iψγ · p) .
(5.3.10b)
The constraint γ ·p = 0 (the Dirac equation) is just a factorized form of the constraint (2.2.8) for this particular representation of the Lorentz group.
A further constraint is necessary to get an irreducible Poincaré representation in
even D. Since any function of an anticommuting coordinate contains bosonic and
5.3. Spinning
89
fermionic terms as the coefficients of even and odd powers of that coordinate, we
need the constraint γD = ±1 on the field (where γD means just the product of all the γ’s) to pick out a field of just one statistics (in this case, a Weyl spinor: notice that D is even in order for the previous constraint to be applied). In the OSp approach
this Weyl chirality condition can also be obtained by an extension of the algebra
[4.10]: OSp(1,1|2)⊗U(1), where the U(1) is chiral transformations, results in an extra Kronecker δ which is just the usual chirality projector. This U(1) generator (for at least the special case of a Dirac spinor or Ramond string) can also be derived as a constraint from first-quantization: The classical mechanics action for a Dirac spinor, under the global transformation δγa ∼ ǫabc···dγbγc · · · γd, varies by a boundary term
∼ dτ ∂ γ
∂τ
D, where as usual γD ∼ ǫabc···dγaγbγc · · · γd. By adding a lagrange multiplier
term for γD ±1, this symmetry becomes a local one, gauged by the lagrange multiplier (as for the other equations of motion). By 1D supersymmetrization, there is also a
lagrange multiplier for ǫabc···dpaγbγc · · · γd. The action then describes a Weyl spinor.
Many supersymmetric gauges are possible for g and ψ. The simplest sets both
to constants (“temporal” gauge G = 1), but this gauge doesn’t allow an OSp(1,1|2)
algebra. The next simplest gauge, dG = 0, does the same to ψ but sets the τ
derivative of g to vanish, making it an extra coordinate in the field theory (related to x−, or p+), giving the generators of (3.4.2). However, the gauge which also keeps ψ as a coordinate (and as a partner to g) is .
G = 0. In order to get the maximal
coordinates (or at least zero-modes, for the string) we choose an OSp(1,1|2) which
keeps ψ (related to ˜
γ±, and the corresponding extra ghost, related to ˜γα). This gives
the modified BRST algebra of (3.5.1).
An “isospinning” particle [5.10] can be described similarly. By dropping the ψ
term in (5.3.6,10b) it’s possible to have a different symmetry on the indices of (γ, ζ) than on those of (x, p). In fact, even the range of the indices and the metric can
be different. Thus, spin separates from orbital angular momentum and becomes
isospin. There is no longer an anticommuting gauge invariance, but with a positive
definite metric on the isospinor indices it’s no longer necessary to have one to maintain unitarity. If we use the constraint ˆ
γ = 0 we get an isospinor, but if we don’t we get a
matrix, with the γ’s acting on one side and the ˆ
γ’s on the other. Noting that τ reversal
switches γ with −ˆγ, we see that the matrix gets transposed. Therefore, the complex conjugation that is the quantum-mechanical analog of τ reversal is actually hermitian conjugation, particularly on a field which is a matrix representation of some group.
(When ˆ
γ is constrained to vanish, τ reversal is not an invariance.) By combining
90
5. PARTICLE
these anticommuting variables with the previous ones we get an isospinning spinning particle.
At this point we take a slight diversion to discuss properties of spinors in arbitrary dimensions with arbitrary spacetime signature. This will complete our discussion of spinors in this section, and will be useful in the following section, where representations of supersymmetry, which is itself described by a spinor generator, will be found to depend qualitatively on the dimension. The analysis of spinors in Euclidean space (i.e, the usual spinor representations of SO(D)) can be obtained by the usual group theoretical methods (see, e.g., [5.11]), using either Dynkin diagrams or an explicit representation of the γ-matrices. The properties of spinors in SO(D+,D−) can then
be obtained by Wick rotation of D− space directions into time ones. (Of particu-
lar interest are the D-dimensional Lorentz group SO(D−1,1) and the D-dimensional
conformal group SO(D,2).) This affects the spinors with respect to only complex
conjugation properties. A useful notation to classify spinors and their properties
is: Denote a fundamental spinor (“spin 1/2”) as ψα, and its hermitian conjugate
as − ¯
ψ.. Denote another spinor ψα such that the contraction ψαψ
α
α is invariant un-
der the group, and its hermitian conjugate ¯
ψ.α. The representation of the group on
these various spinors is then related by taking complex conjugates and inverses of
the matrices representing the group on the fundamental one. For SO(D+,D−) there
are always some of these representations that are equivalent, since SO(2N) has only 2 inequivalent spinor representations and SO(2N+1) just 1. (In γ-matrix language,
the Dirac spinor can be reduced to 2 inequivalent Weyl spinors by projection with
1 (1
2
± γD) in even dimensions.) In cases where there is another fundamental spinor
representation not included in this set, we also introduce a ψα′ and the corresponding 3 other spinors. (However, in that case all 4 in each set will be equivalent, since there are at most 2 inequivalent altogether.) Many properties of the spinor representations can be described by classifying the index structure of: (1) the inequivalent spinors, (2) the bispinor invariant tensors, or “metrics,” which are just the matrices relating the equivalent spinors in the sets of 4, and (3) the σ-matrices (γ-matrices for D odd, but in even D the matrices half as big which remain after Weyl projection), which are simply the Clebsch-Gordan coefficients for relating spinor⊗spinor to vector. In the latter 2 cases, we also classify the symmetry in the 2 spinor indices, where appropriate.
The metrics are of 3 types (along with their complex conjugates and inverses): (1)
.
M β
α , which gives charge conjugation for (pseudo)real representations, and is related to complex conjugation properties of γ-matrices, (2) M
, which is the matrix which
α.
β
5.3. Spinning
91
relates the Dirac spinors Ψ and ¯
Ψ, if it commutes with Weyl projection, and is related
to hermitian conjugation properties of γ-matrices, and (3) Mαβ, which is the Clebsch-Gordan coefficients for spinor⊗same-representation spinor to scalar, and is related to transposition properties of γ-matrices. For all of these it’s important to know
whether the metric is symmetric or antisymmetric; in particular, for the first type we get either real or pseudoreal representations, respectively. In γ-matrix language, this charge conjugation matrix is straightforwardly constructed in the representation where the γ-matrices are expressed in terms of direct products of the Pauli matrices for the 2-dimensional subspaces. Upon Wick rotation of 1 direction each in any
number of pairs corresponding to these 2-dimensional subspaces, the corresponding
Pauli matrix factor in the charge conjugation matrix must be dropped (with perhaps
some change in the choice of Pauli matrix factors for the other subspaces). It then follows that (pseudo)reality is the same in SO(D++1,D−+1) as in SO(D+,D−), so
all cases follow from the Euclidean case. For the second type of metric, ¯
Ψ = Ψ† in
the Euclidean case, so M
is just the identity matrix (i.e., the spinor representations
α.
β
are unitary). After Wick rotation, this matrix becomes the product of all the γ-
matrices in the Wick rotated directions, since those γ-matrices got factors of i in the Wick rotation, and thus need this extra factor to preserve the reality of the tensors
¯
Ψγ · · · γΨ. The symmetry properties of this metric then follow from those of the
γ-matrices. Also, because of the signature of the γ-matrices, it follows that this
metric, except in the Euclidean case, has half its eigenvalues +1 and half −1. The
last type of metric has only undotted indices and thus has nothing to do with complex conjugation, so its properties are unchanged by Wick rotation. It’s identical to the first type in Euclidean space (since the second type is the identity there; in general, if 2 of the metrics exist, the third is just their product), which thus determines it in the general case. Various types of groups are defined by these metrics alone (real, unitary, orthogonal, symplectic, etc.), with the SO(D+,D−) group as a subgroup.
(In fact, these metrics completely determine the SO(D+,D−) group, up to abelian
factors, in D ≡ D+ + D− ≤ 6, and allow all vector indices to be replaced by pairs
of spinor indices. They also determine the group in D = 8 for D− even, due to
“triality,” the discrete symmetry which permutes the vector representation with the 2 spinors.) We also classify the σ-matrices by their symmetry properties only when its 2 spinor indices are for equivalent representations, so they are unrelated to complex conjugation (both indices undotted), and thus their symmetry is determined by the
Euclidean case.
