Odd coset quantum mechanics

The standard quantum states of $n$ complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same `free-fermion' model has a non-linearly realized $SU(n|1)$ symmetry; it can be viewed as the mechanics of a `particle' on the Grassmann-odd coset space $SU(n|1)/U(n)$. We implement a quantization of this model for which the states with non-zero norm transform as a representation of $SU(n|1)$, the representation depending on the U(1) charge of the wave-function. For $n=2$ the wave-function can be interpreted as a BRST superfield.


Introduction
Pseudo-classical mechanics models, with anticommuting variables, have found various applications. One class of applications is to the pseudo-classical description of spin. Consider the 'free-fermion' Lagrangian for n complex anticommuting variables ζ i and their complex conjugatesζ i (γ is a real, positive, dimensionless coupling constant). This Lagrangian has an obvious U(n) invariance but it is also invariant under the larger group SO(2n). In a (coherent state) basis for which the quantum operators corresponding to the variablesζ i are diagonal, with eigenvaluesζ i , the Hilbert space of the quantum theory is spanned by anti-holomorphic functions Φ({ζ}). This space has dimension 2 n and carries a spinor representation of SO(2n).
The full symmetry group is actually much larger than either SO(2n) or SU(n|1); it is the supergroup of supersymplectic diffeomorphisms of a superspace of real dimension (0|2n), which is generated by all 2 2n functions on the Grassmann-odd phasespace. An alternative characterization of it is as the closure of its two subgroups SO(2n) and SU(n|1). Both subgroups contain U(n), which acts in an obvious way on quantum wave-functions, so the Hilbert space decomposes into representations of U(n). In the standard quantum theory these representations combine to yield the spinor of SO(2n) and the Hilbert space norm is the standard scalar product of two spinors.
However, one could attempt to combine the U(n) representations into representations of SU(n|1) rather than SO(2n). In this case, the 'Hilbert' space would be a vector superspace of dimension (2 n−1 |2 n−1 ) rather than a vector space of dimension 2 n , so this quantization of the free-fermion model would be very different from the standard one; a motivation for considering this possibility is that the n = 2 'Hilbert' space would then carry a representation of the Euclidean BRST group SU(2|1) ∼ = OSp(2|2) (see e.g. [1]).
We shall show here that this non-standard quantization can be implemented, but the result depends on the resolution of an operator ordering ambiguity which leads to an ambiguity in the definition of the U(1) charge or, equivalently, the assignment of U(1) charge to the wave-function. There is a 'natural' choice, for which the U(1) charge is the direct quantum analog of the U(1) Noether charge of the Lagrangian (1) but, for completeness, we consider other choices too. In many cases the 'Hilbert' space has zero norm states so the physical states in 'Hilbert' space should be taken to be the equivalence classes of states with non-zero norm modulo the addition of zero norm states. The SU(n|1) representation content of the physical 'Hilbert' space depends on the U(1) charge assigned to the wave-function. For the 'natural' resolution of the operator ordering, and γ = n − 1, we find that the physical Hilbert space is an SU(n|1) singlet! Presumably, these results could be derived by a direct attempt to implement the SU(n|1) symmetry on the 'Hilbert' space found by canonical quantization of (1) but the non-analyticity of the non-linear transformations (2) makes it difficult to see how to do this. We can overcome this problem by introducing the new variables in terms of which the SU(n|1) supersymmetry transformations are analytic: The Lagrangian in these new variables is 1 This Lagrangian can be shown to be the 1-dimensional pullback of the U(1) connection one-form in the nonlinear realization of SU(n|1) in the Grassmann-odd coset space SU(n|1)/U(n). It is shifted by a total derivative under the SU(n|1) transformations and so is a sort of 1-dimensional Wess-Zumino (WZ) term. It contains 'interactions' which complicate the canonical quantization procedure, but this problem is easily solved in a way that will now be described.

Analytic quantization
An equivalent phase-space form of the Lagrangian (5) is where λ i are Lagrange multipliers for n complex phase space constraints, with constraint functions Solving these constraints returns us to the original Lagrangian, up to a total derivative. The SU(n|1)-supersymmetry transformations of the new Lagrangian are The n complex constraint functions {ϕ} are equivalent to 2n real constraint functions that are second class, in Dirac's terminology. However, the n complex constraint functions are in involution; it is only when we consider their complex conjugates that the system of constraints becomes second class. In [3,4] it was shown that when 2n real second class constraints can be separated into two sets of n real constraints in involution then one may quantize without constraints on canonical space variables by imposing one set of n constraints on the Hilbert space states and discarding the other set 2 . Here we shall adopt an 'analytic' version of this procedure which, for Grassman variables, actually preceded the formulation of the method for the real case; in this context it has been called 'Gupta-Bleuler quantization' [6,7].
As the method involves working with an unconstrained phase space, the anticommutation relations follow directly from the canonical Poisson brackets, and these may be realized by setting 3 To take the constraints into account we require that physical states be annihilated by the n operators ϕ i ; this is equivalent to the 'analyticity' conditions 4 on wave-functions Ψ({ξ}, {ξ}). These conditions have the solution for anti-analytic Φ, which has the expansion where In principle each of the 2 n coefficients could have any Grassmann parity but to implement SO(2n) invariance we would have to choose all of them to have the same Grassmann parity, which must be even for a positive definite norm. In this way we would recover the standard free-fermion Hilbert space, as a 2 n -dimensional vector space, although the SO(2n) invariance is not manifest in our approach and has to be imposed. Here however, we wish to explore the alternative possibility that the 'Hilbert' space carries some representation of the supergroup SU(n|1) . For this to be possible we must take the anti-analytic function Φ to have a definite Grassmann parity 5 . In this case the 'Hilbert' space is a vector superspace of dimension (2 n−1 |2 n−1 ) ; for a reason to be made clear later, we assume that Φ is Grassmann-even for n even and Grassmann-odd for n odd. Our next task is to determine how SU(n|1) acts in this 'Hilbert' space.

SU (n|1) in 'Hilbert' space
The linear U(n) transformations of the canonical variables of the Lagrangian (6) are generated by the Noether charges The corresponding quantum U(n) generators are the differential operatorŝ For wave-functions of the form (11) we havê from which we deduce the U(n) transformation of Φ to be The same requirement is made in the standard quantization of the superparticle, in contrast to the 'spinning particle'. In fact, as the 4-dimensional spinning particle and superparticle Lagrangians can be shown to be classically equivalent [11,12,13], the difference between the two can be ascribed to different quantization procedures, in close analogy to the 'free fermion' model considered here.
The nonlinear supersymmetry transformations of (8) are generated by the Grassmannodd Noether charges Note the presence of the terms linear in ξ andξ; these arise from the fact that the supersymmetry variation of the Lagrangian is not zero but rather a total time derivative. These terms have no effect on the transformations of ξ i generated bŷ S andŜ, which are those of (4), but they do contribute to the U(1) charge in the SU(n|1) superalgebra of Poisson brackets of Noether charges. In fact, one finds that the U(1) charge is where the shift by γ is directly attributable to the γ-dependent linear terms inŜ andŜ. In passing to the quantum theory, the coefficients of these terms become ambiguous because of operator ordering ambiguities. This ambiguity is partially fixed by requiring that physical wave-functions Ψ of the form (11) transform into physical wave-functions, i.e.
This leaves us with the following quantum supersymmetry generators, parametrized by a real number α:Ŝ These have the anticommutation relation whereB is the quantum U(1) generator One sees from this that the choice α = γ is 'natural' because it leads to a quantum U(1) ⊂ SU(n|1) generator that is the direct quantum counterpart of the classical U(1) charge B of (19). Nevertheless, we shall consider the case of general α in what follows.
We now compute the action of the chargesŜ i ,Ŝ i , on physical wave-functions. One finds thatŜ These results yield the following SU(n|1)-supersymmetry transformation of Φ: For component fields in the expansion (12) this transformation implies The full set of SU(n|1) transformations of Φ are such that where s is a local function ofξ. Thus, Φ is a scalar anti-analytic superfield when q=0; for other values of q, including the 'natural' value q = γ, one may consider Φ as a charged scalar superfield, with charge q.
We have supposed up to now that α and γ are arbitrary real variables but one might expect the combination q to be quantized 6 . As we shall see, the representation content of the physical 'Hilbert' space depends on q and simplifications, associated with the existence of zero norm states, occur for special integer values of q.

SU (n|1)-invariant norm
In order to construct an invariant inner product one must first obtain an SU(n|1) invariant measure under the coordinate transformations (4). With the help of the lemma δ ǫ 1 +ξ · ξ = ǭ · ξ − ǫ ·ξ 1 +ξ · ξ , it is not difficult to show that the SU(n|1) invariant measure is such that where However, because the transformation (26) involves a U(1) weight term, an additional factor is needed in the measure when q = 0. Let us replace Φ by Φ (q) to remind us that Φ carries U(1) charge q. Then the following bilinear form is SU(n|1) invariant: Note that the additional factor in the measure is unity precisely when q = 0 but is non-trivial otherwise.
In terms of the original wave-functions Ψ = 1 +ξ · ξ − γ 2 Φ (q) , the SU(n|1) invariant scalar product corresponding to the definition (33) reads and Ω(ξ,ξ) is another vector in the same 'Hilbert space'. It is straightforward to check that the quantum generators (21) are mutually conjugate with respect to this scalar product On the other hand, for κ = 0 the fermionic momentum operators ∂/∂ξ i and ∂/∂ξ i are not mutually conjugate with respect to (34). Note, however, that (34) is defined modulo the following similarity transformation (change of basis) in 'Hilbert space' This amounts to the substitution Ω, Ψ → Ω ′ , Ψ ′ and shift κ → κ+ 2λ in the definition (34), as well as a corresponding change in the observables. The conjugacy property (36) of the SU(n|1) supersymmetry generators is evidently basis-independent. In contrast, an analogous conjugacy property holds for the fermionic momentum operators only for the special choice of basis corresponding to λ = −κ/2: Thus, the fermionic momentum operators (9) are mutually conjugate in the sense that there is a basis in 'Hilbert space' for which they satisfy (38).
Let us now turn to the analysis of the field content of Φ implied by the invariant norm (33). In general, there are contributions to (33) from all coefficients in the expansion (12), but zero norm states occur for special values of q. For example, when q = n − 1 we have As δ ǫ c (n) = 0 for this choice of q we see that the physical Hilbert space is an SU(n|1) singlet. All functions Φ (n−1) with c (n) = 0 have zero norm. If instead we set q = n − 2 then we find Again there are zero norm states because the SU(n) representation content appearing in the norm is restricted to n ⊕ 1; these SU(n) representations combine to form the fundamental n + 1 representation of SU(n|1). As a final example, consider q = 0. In this case we have The SU(n) representation content is n ⊕ n(n − 1)/2 ⊕ . . . ⊕ n ⊕ 1.
An inspection of the transformations (28) leads to the following general conclusions about the structure of the 'Hilbert spaces' corresponding to different choices of q. For the choice .
(43) Fork = 0 this subspace is the full space of coefficients of Φ but otherwise it is not, and the remaining coefficients are transformed into the above set; this shows that the representation of SU(n|1) carried by Φ (k−1) is reducible but not fully reducible. The norm ||Φ (k−1) || includes only the components (43), so there exist zero norm states unlessk = 0. As the set (43) is irreducible under the action of SU(n|1), we can consistently set them to zero: c The complementary set of coefficients then forms an irreducible set on its own, and one would expect there to exist a corresponding SU(n|1)-invariant norm. However, the 'obvious' norm, defined by (33), is identically zero when (44) is satisfied; this is easily seen by rewriting (44) in the superfield form ∂kΦ (q) ∂ξ i 1 . . . ∂ξ ik = 0 (and c.c.) .
One can check that these constraints are covariant under (26) provided that the condition (42) holds. Of course, they do not correspond to constraints in the Lagrangian (6) so what we are now doing cannot be considered as a quantization of that Lagrangian but one could add to it the classical constraints corresponding to (45), for which the constraint functions are polynomials inπ. It is remarkable that for Φ (k−1) constrained by (45) there exists the following alternative norm Taking into account that it is straightforward to prove invariance of (46) given the constraints (45), which are crucial to the result. It is interesting that the 'Lagrangian density' in (46) is not a tensor one as in (33), but has an additional variation into a total derivative, as is typical for WZ or Chern-Simons lagrangians.

n = 2 and BRST
We shall now illustrate the above results with the n = 2 case; we also choose γ = 1, which means that the 'natural' choice of operator ordering corresponds to q = 1. For n = 2 we can interpret the odd coset space SU(2|1)/U(2) as a BRST superspace because SU(2|1) ∼ = OSp(2|2) is the Euclidean BRST supergroup. For n = 2 we have The coefficients (b 1 , b 2 ), which form an SU(2) doublet, can be interpreted as (euclidean) Faddeev-Popov ghost and antighost for the SU(2)-singlet gauge-fixing term a; the other SU(2) singlet c is then the 'Nakanishi-Lautrup' auxiliary field. ¿From (28) we deduce that the SU(2|1)-supersymmetry transformations for q = 0 are where ε 12 = ε 12 = 1, ε ik ε il = δ k l . This is not a reducible representation because a transforms non-trivially while (b 1 , b 2 , c) span a 3-dimensional invariant subspace. Observe that is invariant. Of course, this is not really a norm as the variables b i are anticommuting. 7 In other words, the physical states are vectors in a vector superspace of dimension (1|2) transforming as the fundamental representation of SU(2|1) . For q = 1 the transformations (49) become This is the 'natural' case for which the physical Hilbert space is a singlet. Indeed, the norm (33) in this case is simply Still with q = 1, we may impose the covariant condition This leaves us with the irreducible multiplet (a, b i ): The alternative norm is so physical states once again transform as a fundamental (1 + 2) representation of SU(2|1) (the precise correspondence with the realization (49), (50) is achieved via substitutions a →c, b i → ǫ ikb k , wherec andb i are transformed just as c and b i ). Finally, we shall consider q = −1, for which the transformation law (26) becomes and the invariant norm calculated by the formula (33) is In this case one cannot single out any invariant subspace and so ends up with a 4-dimensional irreducible multiplet (b 1 , b 2 , c, a) of SU(2|1).

Discussion
We have shown that the mechanics of n free complex Grassmann-odd variables provides a non-linear realization of the supergroup SU(n|1). It can be viewed as the mechanics of a 'particle' with the Grassmann-odd coset space SU(n|1)/U(n) as its phase space. This model is trivial in the sense that its Hamiltonian vanishes but it is the simplest of a class of models that realize SU(n|1) non-linearly and for which the Hamiltonian is generically non-zero. The particle on SU(2|1)/[U(1) × U(1)] is an example, and one that will be considered in a future publication. Part of the motivation of this paper was to exhibit some of the properties of these models in the simplest possible setting. Another motivation is that coset-spaces of the n = 2 supergroup SU(2|1) can be interpreted as BRST superspaces. We have shown that there exists an alternative quantization of the 'free fermion' model that implements the classical SU(n|1) symmetry. In contrast to the standard quantization, for which the states transform as a spinor of SO(2n), the states of the alternative quantum theory are vectors in a vector superspace transforming under SU(n|1). The specific SU(n|1) representation content depends on the resolution of an operator ordering ambiguity, which amounts to a choice of U(1) charge for the wave-function. There is a natural choice, given the initial classical Lagrangian, because this Lagrangian can be viewed as a WZ term for U(1) ⊂ SU(n|1) and this leads to specific shift in the U(1) generator that is naturally identified with the U(1) charge of the quantum wave-function. For this choice, and a particular choice of the 'coupling constant', the 'Hilbert' space contains zero norm states and the physical 'Hilbert' space is an SU(n|1) singlet.
For other choices of U(1) charge assignment (and other choices of coupling constant) one gets other representations of SU(n|1), picked out by an SU(n|1) invariant norm. We showed that there exists a class of integer U(1) charge assignments for which the representation is irreducible. Remarkably, in this case the complementary representation contained in the wave-function, again irreducible, could be picked out by a different invariant, but not manifestly-invariant, norm provided that the original representation was constrained to be absent; this case corresponds to the quantization of the original free-fermion Lagrangian with additional phase space constraints.
Any quantization of Grassmann-odd variables has to take into account (explicitly or implicitly) second-class phase-space constraints. In our case these were non-trivial because of a redefinition of variables needed for analyticity of the SU(n|1) transformations. We dealt with these constraints by a variant of the 'gauge unfixing' method involving a separation of the constraints into analytic and anti-analytic subsets in involution. It may be helpful if we sketch here how this method can be used to covariantly quantize the massless 4D superparticle, as done in [8,9]. The fermionic constraint operators are the supercovariant derivatives D α and their complex conju-gatesDα. These are not all second class (given p 2 = 0) because the combinations p ααDα and p αα D α are first class. Although we should require that both of these first class operators annihilate physical states Φ we need only impose p αα D α Φ = 0 explicitly if we also imposeDαΦ = 0, as required for 'analytic quantization', because the other one is then implied. The independent constraints are thereforeDαΦ = 0 and p αα D α Φ = 0 (because these imply p 2 = 0), but these are just the free field equations for a massless chiral superfield.