BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

Nilpotency ofQ BRST and background field equations

Following the method of Fradkin and Vialkovsky, we study the BRST charge operator.Q _BRST , for the closed bosonic string coupled to the background of its massless states. We show that the requirement of nilpotency ofQ _BRST gives rise to the correct equations for the background fields.     

All free string theories are theories of BRST cohomology

The fully gauge-invariant theory of the free open (super) strings is generalized to treat closed (super) strings in the BRST formalism. It is shown that the actions for all theories can be written in the form (Φ,Q_BΦ) using the BRST charge Q_B and the string functional Φ, if the inner product is interpreted appropriately with necessary ghost insertions. Also is shown that consistent truncations of our actions recover all known covariant formulations of (super) strings.     

Canonical BRST quantisation of worldsheet gravities

We reformulate the BRST quantisation of chiral Virasoro and W₃ worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator Q, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator Q², calculated using operator products, to the gauge fermion.     

N-Superstring vertex and the vertex operator for the emission of the superstring

The BRST invariant, supersymmetric N-string vertex which applies to both, the Neveu-Schwarz and Ramond sector of the superstring is formulated using the vertex operator for the emission of a superstring. It is shown that the N-superstring vertex thus obtained is cyclic symmetric when GSO projected on-shell states and operators for the external strings are applied. The constraint equations of this vertex and their singularity structure are examined and we show that this vertex also has the required property of a transition operator. We also give a proof of its BRST invariance and supersymmetry.     

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?^* _ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?^* _ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables. Received: 28 April 2000 / Accepted: 6 December 2000     

On the associativity anomaly in open string field theory

The associativity anomaly in the star algebra of the open bosonic string is demonstrated by a simple oscillator calculation. In an associative algebra, formal arguments require that the “half string” BRST operator Q_L be nilpotent; however, we show that associativity is actually violated by computing Q_L² in an explicit operator representation of the star algebra.     

Square-roots,internal symmetries,and fermion field quantization

It is observed that the complex square-root of the hermitian matrix σ^μ p _μ associated with a physical four-momentum admits Lorentz-independent unitary transformations that may be related to the internal symmetries of hadrons. An operator-valued square-root of the Hilbert space inner product in relativistic one-electron theory brings in fermion field quantization conditions as a direct concomitant of its linearity and hermicity properties.     

A new finite-dimensional pair coherent state studied by virtue of the entangled state representation and its statistical behavior

In this paper we construct a new type of finite-dimensional pair coherent states |ξ, q〉 as realizations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, the nonorthogonality and completeness properties of the state |ξ, q〉 are investigated. Based on the Wigner operator in the entangled state |τ〉 representation, the Wigner function of |ξ, q〉 is obtained. The properties of |ξ, q〉 are discussed in terms of the negativity of its Wigner function. The tomogram of |ξ, q〉 is calculated with the aid of the Radon transform between the Wigner operator and the projection operator of the entangled state |η, κ₁, κ₂〉. In addition, using the entangled state |τ〉 representation of |ξ, q〉 to show that the states |ξ, q〉 are just a set of energy eigenstates of time-independent two coupled oscillators.     

BRST invariant tadpole operator in the bosonic string

Using a BRST invariant three-reggeon vertex we construct an explicit expression for the BRST invariant planar tadpole operator for the bosonic string. Joined to an (N+1)-reggeon vertex it yields the correct expression for the one-loop N-tachyon amplitude.     

Rotational motion,shell structure and coherent state dynamics

We analyze the dynamics of coherent states in a two-dimensional model of rotational motion using the unitary group and the schematic Q·Q force. We find a uniform rotation of the nuclear quadrupole. Application of the gauge invariant periodic quantization method yields the exact rotational spectrum.     

BRST invariant mixed string vertex for the bosonic string

We construct a BRST invariant (N + M)-string vertex including both open and closed string states. When we saturate it with N open string and M closed string physical states it reproduces their corresponding scattering amplitude. As a particular case we obtain a BRST invariant vertex for the open-closed string transition.     

Covariant quantization of string based on BRS invariance

Covariant canonical quantization of the bosonic string is performed, based on the BRS invariance principle. Defining a physical state by a modified form of Kugo-Ojima's subsidiary condition Q_B|phys〉 = 0 (Q_B = BRS charge), we prove a no-ghost theorem in a simple manner. There an interesting relation between critical dimension and BRS symmetry is noticed; namely, the conditions D = 26 and α₀ = 1 for the space-time dimension D and the zero-intercept α₀ of leading trajectory are required by the nilpotency Q_B² = 0 of the BRS charge. In addition, ghost number fractionization is observed in this system.     

BRST quantization of the superparticle

We show that the quantization of the superparticle action is possible. This is done by shifts in the BRST operator and the resulting action has an infinite number of ghosts. The total BRST operator is given by an infinite sum and is shown to be nilpotent. We also obtain a BRST invariant kinetic operator that contains the dynamical, auxiliary and gauge pieces in it.     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

The algebraic structure of the fermionic string

The local structure of the product expansion algebra of the covariant NSR string is analyzed. An “on-shell” Kac-Moody like algebra is found to generate the BRST invariant part of the covariant lattice Γ_5,1. This algebra is a local version of ten-dimensional SUSY.     

String loop effect on the BRST charge

An effective BRST charge Q_BRST which incorporates the string one-loop corrections is presented for the closed basonic string in an arbitrary background. The effective σ-model action which leads to such a Q_BRST is obtained and some consequences are discussed.     

Orthosymplectic unification of the Galilei group and BRST

The inclusion of BRST generators into the Poincarè group in D dimensions is known to lead IOsp[D,2|2]. Similarly, conformal symmetry gets extended into Osp[D×1,3|2]. For the non-relativistic case we find that the Galilei symmetry gets extended, by inclusion of the BRST generators, into an orthosymplectic symmetry possessing Osp[D,1|2] as a subgroup. All such extensions express the possibility of formulating the classical theories in reparametrization invariant ways. They include besides the generators of the initial kinematical symmetry (Poincarè, or conformal, or Galilei), the generators of Parisi-Sourlas transformations. The extended symmetries follow directly through BRST quantization.     

BRST operator on a Riemann surface

We discuss construction of the BRST operator for the bosonic closed string theory on an arbitrary Riemann surface of genus g>1. Using the BRST operator we prove the decoupling of the bosonic spurious physical states in general.     

On BRST quantization of second class constraint algebras

A BRST quantization of second-class constraint algebras that avoids Dirac brackets is constructed, and the BRST operator is shown to be related to the BRST operator of first class algebra by a nonunitary canonical transformation. The transformation converts the second class algebra into an effective first class algebra with the help of an auxiliary second class algebra constructed from the dynamical Lagrange multipliers of the Dirac approach. The BRST invariant path integral for second class algebras is related to the path integral of the pertinent Dirac brackets, using the Parisi-Sourlas mechaism. As an application the possibility of string theories in subcritical dimensions is considered.