The Parastatistics Fock Space and Explicit Infinite-Dimensional Representations of the Lie Superalgebra $${\mathfrak {osp}}(2m+1|2n)$$ osp ( 2 m + 1 | 2 n )

2016 ◽  
pp. 169-180
Author(s):  
N. I. Stoilova ◽  
J. Van der Jeugt
Keyword(s):  
Fock Space ◽  
Lie Superalgebra ◽  
Infinite Dimensional

scholarly journals Semi-device-independent framework based on natural physical assumptions

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 33 ◽  
Author(s):  
Thomas Van Himbeeck ◽  
Erik Woodhead ◽  
Nicolas J. Cerf ◽  
Raúl García-Patrón ◽  
Stefano Pironio
Keyword(s):  
Fock Space ◽  
Outcome Measurement ◽  
Photon Number ◽  
Quantum Correlations ◽  
Mean Value ◽  
Laser Source ◽  
Optical Scheme ◽  
Infinite Dimensional ◽  
Energy Constrained ◽  
Quantum Optical

The semi-device-independent approach provides a framework for prepare-and-measure quantum protocols using devices whose behavior must not be characterized nor trusted, except for a single assumption on the dimension of the Hilbert space characterizing the quantum carriers. Here, we propose instead to constrain the quantum carriers through a bound on the mean value of a well-chosen observable. This modified assumption is physically better motivated than a dimension bound and closer to the description of actual experiments. In particular, we consider quantum optical schemes where the source emits quantum states described in an infinite-dimensional Fock space and model our assumption as an upper bound on the average photon number in the emitted states. We characterize the set of correlations that may be exhibited in the simplest possible scenario compatible with our new framework, based on two energy-constrained state preparations and a two-outcome measurement. Interestingly, we uncover the existence of quantum correlations exceeding the set of classical correlations that can be produced by devices behaving in a purely pre-determined fashion (possibly including shared randomness). This feature suggests immediate applications to certified randomness generation. Along this line, we analyze the achievable correlations in several prepare-and-measure optical schemes with a mean photon number constraint and demonstrate that they allow for the generation of certified randomness. Our simplest optical scheme works by the on-off keying of an attenuated laser source followed by photocounting. It opens the path to more sophisticated energy-constrained semi-device-independent quantum cryptography protocols, such as quantum key distribution.

scholarly journals White noise delta functions and continuous version theorem

1993 ◽  
Vol 129 ◽  
pp. 1-22
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Harmonic Analysis ◽  
Quantum Physics ◽  
Fock Space ◽  
Continuous Version ◽  
Infinite Dimensional ◽  
Schwartz Distribution ◽  
Delta Functions ◽  
Gelfand Triple ◽  
Gaussian Space

The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

scholarly journals A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Asao Arai
Keyword(s):  
Dirac Operator ◽  
Hilbert Spaces ◽  
Special Class ◽  
Spectral Properties ◽  
Fock Space ◽  
Dirac Operators ◽  
Fredholm Property ◽  
Perturbation Parameter ◽  
Coupling Constant ◽  
Infinite Dimensional

Spectral properties of a special class of infinite dimensional Dirac operatorsQ(α)on the abstract boson-fermion Fock spaceℱ(ℋ,𝒦)associated with the pair(ℋ,𝒦)of complex Hilbert spaces are investigated, whereα∈Cis a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operatorQ(0)is taken to be a free infinite dimensional Dirac operator. A variety of the kernel ofQ(α)is shown. It is proved that there are cases where, for all sufficiently large|α|withα<0,Q(α)has infinitely many nonzero eigenvalues even ifQ(0)has no nonzero eigenvalues. Also Fredholm property ofQ(α)restricted to a subspace ofℱ(ℋ,𝒦)is discussed.

scholarly journals PARAFERMIONS, PARABOSONS AND REPRESENTATIONS OF 𝔰𝔬(∞) AND 𝔬𝔰𝔭(1|∞)

2009 ◽  
Vol 20 (06) ◽  
pp. 693-715 ◽  
Author(s):  
N. I. STOILOVA ◽  
J. VAN DER JEUGT
Keyword(s):  
Lie Algebra ◽  
Fock Space ◽  
Complete Solution ◽  
Lie Superalgebra ◽  
Explicit Construction ◽  
Fock Spaces ◽  
Infinite Rank ◽  
Infinite Set ◽  
Basis Vectors

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra 𝔰𝔬(∞) and of the Lie superalgebra 𝔬𝔰𝔭(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand–Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations.

QUANTIZATION AND COCYCLES ON THE SUPERTORUS AND LARGE-N LIMITS FOR THE CLASSICAL LIE SUPERALGEBRAS

1991 ◽  
Vol 06 (03) ◽  
pp. 217-224 ◽  
Author(s):  
E.S. FRADKIN ◽  
V. Ya. LINETSKY
Keyword(s):  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Large N

The Poisson superbracket Lie superalgebra on the supertorus T2d|N is considered and its quantization is carried out. It is shown that there exists a non-trivial supercentral extension by means of 2d arbitrary c-numbers (when N is even), or 2d Grassmann numbers (when N is odd). It is shown that the infinite-dimensional superalgebras on the supertorus T2d|N can be considered as certain generalizations and large-M limits of the classical superalgebras A(M| M) and Q(M) (when N is even and odd respectively).

scholarly journals Functional Representations for Fock Superalgebras

1998 ◽  
Vol 01 (02) ◽  
pp. 285-324 ◽  
Author(s):  
Joachim Kupsch ◽  
Oleg G. Smolyanov
Keyword(s):  
Reproducing Kernel ◽  
Fock Space ◽  
Function Spaces ◽  
Graded Algebras ◽  
Fock Representation ◽  
Infinite Dimensional ◽  
Gaussian Integration ◽  
Classical Function ◽  
Functional Representations ◽  
Wick Ordering

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.

scholarly journals Hardy-Type Space Associated with an Infinite-Dimensional Unitary Matrix Group

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Oleh Lopushansky
Keyword(s):  
Fock Space ◽  
Matrix Group ◽  
Unitary Matrix ◽  
Type Space ◽  
Infinite Dimensional ◽  
System A ◽  
Hardy Type ◽  
Complex Functions ◽  
Hardy Type Space ◽  
The Right

We investigate an orthogonal system of the homogenous Hilbert-Schmidt polynomials with respect to a probability measure which is invariant under the right action of an infinite-dimensional unitary matrix group. With the help of this system, a corresponding Hardy-type space of square-integrable complex functions is described. An antilinear isomorphism between the Hardy-type space and an associated symmetric Fock space is established.

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