Fock Vacuum Research Articles

COMMENTS ON HAMILTONIAN FORMALISM OF AdS/CFT CORRESPONDENCE

As a toy model to search for Hamiltonian formalism of the AdS/CFT correspondence, we examine a Hamiltonian formulation of the AdS2/CFT1 correspondence emphasizing unitary representation theory of the symmetry. In the course of a canonical quantization of the bulk scalars, a particular isomorphism between the unitary irreducible representations in the bulk and boundary theories is found. This isomorphism defines the correspondence of field operators. It states that field operators of the bulk theory are field operators of the boundary theory by taking their boundary values in a specific way. The Euclidean continuation provides an operator formulation on the hyperbolic coordinates system. The associated Fock vacuum of the bulk theory is located at the boundary, thereby identified with the boundary CFT vacuum. The correspondence is interpreted as a simple mapping of the field operators acting on this unique vacuum. Generalization to higher dimensions is speculated.

Effect of zero modes on the bound-state spectrum in light-cone quantisation

We study the role of bosonic zero modes in light-cone quantisation on the invariant mass spectrum for the simplified setting of two-dimensional SU(2) Yang-Mills theory coupled to massive scalar adjoint matter. Specifically, we use discretised light-cone quantisation where the momentum modes become discrete. Two types of zero momentum mode appear – constrained and dynamical zero modes. In fact only the latter type of modes turn out to mix with the Fock vacuum. Omission of the constrained modes leads to the dynamical zero modes being controlled by an infinite square-well potential. We find that taking into account the wavefunctions for these modes in the computation of the full bound state spectrum of the two dimensional theory leads to 21% shifts in the masses of the lowest lying states.

Photoemissive sources and quantum stochastic calculus

Just at the beginning of quantum stochastic calculus (QSC), Hudson and Parthasarathy proposed a quantum stochastic Schrodinger equation linked to dilations of quantum dynamical semigroups [7, 8, 11]. Such an equation has found applications in physics, mainly in quantum optics, but not in its full generality [6, 1, 2]. It has been used to give, at least approximately, the dynamics of photoemissive sources such as an atom absorbing and emitting light or matter in an optical cavity, which exchanges light with the surrounding free space. But in these cases the possibility of introducing the gauge (or number) process in the dynamical equation has not been considered. In this paper we want to show, in the case of the simplest photoemissive source, namely a two–level atom stimulated by a laser, how the full Hudson– Parthasarathy equation allows to describe in a consistent way not only absorption and emission, but also the scattering of the light by the atom. Let us recall the Hudson–Parthasarathy equation; this is just to fix our notations, while for the proper mathematical definitions and the rules of QSC we refer to the book by Parthasarathy [11]. We denote by F := F(X ) the Boson Fock space over the Hilbert space X := Z ⊗ L(R+) ≃ L(R+;Z), where Z is another separable complex Hilbert space. Let {ei, i ≥ 1} be a c.o.n.s. in Z and let us denote by Ai(t), A†i (t), Λij(t) the annihilation, creation and gauge processes associated with such a c.o.n.s. We denote by E(h), h ∈ X , the exponential vectors in F with normalization ‖E(h)‖2 = exp{‖h‖2}; E(0) is the Fock vacuum. We shall also use the Boson Fock spaces Ft := F ( L([0, t];Z) ) and F t := F ( L2((t,∞);Z) ) , for which we have F = Ft ⊗F , and the Weyl operators

Wick products of the CAR algebra

The purpose of this paper is to put in a precise mathematical (algebraic) form the Wick products of the CAR algebra. We state in detail the reduction of the ordinary product of Fermi fields in terms of a finite sum of monomials in the creation and annihilation operators in which all creation operators occur to the left of all annihilation operators (Wick-ordered) and the Fock (vacuum) state of the former.

Non-trivial Fock vacuum of the light-front Schwinger model

A Fock representation for the physical vacuum based on the dynamical zero mode of the gauge field is suggested for the Schwinger model quantized on the light front. The θ-vacuum in terms of a gauge invariant superposition of zero-mode coherent states is shown to reproduce the vacuum-angle dependence of the fermion condensate within the bosonized formulation of the model.

Spontaneous Symmetry Breaking in Discretized Light-Cone Quantization

Spontaneous symmetry breaking of the light-front Gross-Neveu model is studied in the framework of the discretized light-cone quantization. Introducing a scalar auxiliary field and adding its kinetic term, we obtain a constraint on the longitudinal zero mode of the scalar field. This zero-mode constraint is solved by using the $1/N$ expansion. In the leading order, we find a nontrivial solution which gives the fermion nonzero mass and thus breaks the discrete symmetry of the model. It is essential for obtaining the nontrivial solution to treat adequately an infrared divergence which appears in the continuum limit. We also discuss the constituent picture of the model. The Fock vacuum is trivial and an eigenstate of the light-cone Hamiltonian. In the large $N$ limit, the Hamiltonian consists of the kinetic term of the fermion with dressed mass and the interaction term of these fermions.

Fermionic Gaussians

The notion of a Gaussian as the exponential of a quadratic is rather familiar. Such functions are of considerable importance in a number of contexts, for example within quantum theory. Thus, in the Schrödinger representation of the canonical commutation relations they alone minimize uncertainty and they appear as ground states for harmonic oscillators. Also in the complex-wave representation of a free boson field they arise as transforms of the Fock vacuum under certain Bogoliubov automorphisms.

Cosmological quantum fluctuations: gauge-invariance and Gaussian states

A quantum theory of cosmological perturbations is based on the action of Einstein gravity with matter expanded to quadratic order around a background solution of the field equations. If, in the Schrodinger picture, the wavefunction is assumed to factorize such that each factor is a function of only a finite number of degrees of freedom ( e.g. Fourier components of the fields) all gauge constraints can be solved explicitly. This generalizes previous gauge-invariant quantizations while clarifying their limited scope. The result is cast into the more convenient Heisenberg picture. It can be found directly from the Lagrangian, the Hamiltonian approach serving only as a justification. The concept of a Fock vacuum is generalized to several linear but coupled degrees of freedom.

Vacuum structures in Hamiltonian light-front dynamics

Hamiltonian light-front dynamics of quantum fields may provide a useful approach to systematic nonperturbative approximations to quantum field theories. We investigate inequivalent Hilbert-space representations of the light-front field algebra in which the stability group of the light front is implemented by unitary transformations. The Hilbert space representation of states is generated by the operator algebra from the vacuum state. There is a large class of vacuum states besides the Fock vacuum which meet all the invariance requirements. The light-front Hamiltonian must annihilate the vacuum and have a positive spectrum. We exhibit relations of the Hamiltonian to the nontrivial vacuum structure.

Quantum white noises—White noise approach to quantum stochastic calculus

Let H = L2 (R) be the Hilbert space of all complex-valued square integrable functions defined on R, Ф = Γ(H) be the Boson Fock space over H. For each h ∈ H, denote by ε(h) the corresponding exponential vector:in particular ε(0) is the Fock vacuum.

PHASE HOLONOMY OF THE VACUUM IN THREE-DIMENSIONAL GAUGE THEORIES

The phase two-form of Berry in the neighborhood of a degeneracy of the Fock vacuum of a semisimple, nonabelian, second-quantized, relativistic fermion-background gauge field Hamiltonian is shown to be that of the Dirac magnetic monopole—thus extending a result of Berry to field theory. The Dirac Hamiltonian for an SU(2) fermion on the two-sphere is solved in a particular two-parameter family of background instanton gauge potentials as an explicit illustrative example.

U q covariant oscillators and vertex operators

U q (sl(2)) covariant Bose and Fermi oscillators are studied systematically and applied to constructing vertex operators and Uq invariant blocks. For qp=−1 (p=2,3,...) a p2-dimensional unitary Fock space representation of the Bose oscillator algebra 𝒜−(q) is constructed. This representation does not allow, however, for the construction of all Uq invariant blocks associated with an indecomposable representation of Uq. An extension of 𝒜− is needed in this case, which involves operators creating states of q spin projection ±p/2 from the Fock vacuum. The extended algebra only has infinite-dimensional indefinite metric space representations. An algebraic notion of trace is introduced (provided q2≠1 in the Bose case). This allows the introduction of a family of (in general, mixed) Uq invariant ‘‘temperature’’ states, such that the vacuum expectation value appears in the zero temperature limit.

Fock space formulation of Chern-Simons theories

A Fock space operator formulation of Chern-Simons-Witten theories is presented, and its connection with the usual holomorphic quantization is established. The conformal blocks are obtained as projections of suitable states on a Fock vacuum. We study in detail the abelian case and sketch its extension to non-abelian groups where the powerfulness of the formalism to compute the corresponding WZW conformal blocks on arbitrary Riemann surfaces is emphasized.

Quasi-coherent states and the spectral resolution of the q-Bose field operator

The single mode q-Bose field for -1<or=q<1 is shown to be a bounded operator with a continuous spectrum lying in the interval between +or-(2/(1-q))12/, thus becoming unbounded in the Bose limit (q to 1). The generalized eigenstates are determined in terms of q-Hermite polynomials whose properties are exploited to obtain orthonormality relations. It is observed that these are connected to important results in combinatorics. It is shown that the eigenstates are products of two q-exponentials of the q-creation operator acting on the Fock vacuum, in contrast to ordinary coherent states which involve only a single q-exponential of the creation operator. A method of representing operators in normal-ordered form based on this representation is described. The spectral resolution is employed to compute the asymptotic temporal behaviour of the particle number expectation value in states produced by coupling the q-Bose field to a source. It is shown that the limit does not commute with the limit q to 1. Specifically it is shown that for O<or=q<1 the asymptotic growth is linear in time in contrast to the quadratic growth for q=1.

Particle picture in external fields, quantum holonomy, and gauge anomaly

We formulate a path integral of chiral gauge theories by means of the canonical quantization of fermions in time-dependent background gauge fields. The expression of the path integral is composed of two parts. One is due to the nontrivial holonomy of the fermionic Fock vacua and the other is the conventional form which is used in the perturbation theory. The nontrivial holonomy part is expected to be a nonlocal counter term. We show a possibility of the perturbative calculation

Functional representation for fermionic quantum fields.

A functional representation for fermionic quantum fields is developed in analogy to familiar results for bosonic fields. The infinite Clifford algebra of the field anticommutator is realized reducibly on a Grassmann functional space. On this space, transformation groups may be represented without normal ordering with respect to a Fock vacuum, and a projective representation for the two-dimensional conformal group is found, which is compared to the corresponding representation in terms of bosonic fields. When a quadratic Hamiltonian for the Fermi fields is posited, a Fock space can be constructed after a prescription for filling the Dirac sea is selected. Different filling prescriptions lead to inequivalent Fock spaces within the functional space. Explicit eigenfunctionals exhibit the peculiarities of fermionic field theory, such as fractional charge, Berry's phase, and anomalies.

Klein Gordon Equations on a Time Lattice

AbstractWe consider the second quantization procedure for a KLEIN GORDON equation with time dependent Hamiltonian and with replaced second order time derivative by the appropriate difference operator. With each time step there is a Bogoljubov transformation which describes particle creation and annihilation and the accompanied change of the Fock vacuum.

Massless minimally coupled scalar field in de Sitter space.

We quantize the massless minimally coupled scalar field in de Sitter space, and find a one-complex-parameter family of O(4)-invariant Hadamard Fock vacua which break de Sitter invariance. The different Fock spaces corresponding to the different choices of vacuum are subspaces of a single space of states. We make some remarks about the existence of E(3)- and O(1,3)-invariant Fock vacuum states.

Closed strings in open-string field theory.

An explicit construction of closed-string states is given in terms of open-string oscillators acting on the open-string Fock vacuum. The closed-open-open coupling is computed for the graviton and agrees with the standard result. The closed string states are not annihilated by the Becchi-Rouet-Stora-Tyutin operator Q. Instead they generate inner derivations which anticommute with Q. This is sufficient to insure gauge invariance of tree-level S-matrix elements. It also implies that the closed-string states obey, level by level, the linearized equation of motion of the cubic action.

Induced fractional spin and statistics in three-dimensional QED

It is shown that in background magnetic fields the fermionic Fock vacuum of (2+1)-dimensional quantum electrodynamics is a gauge invariant eigenstate of the full angular-momentum operator. The eigenvalue is computed and its relation to fractional statistics is discussed. It is argued that anomalous induced angular momentum persists to all orders in perturbation theory.