Fock space


The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space and second quantization").[1][2]

Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the n-particle states are vectors in a symmetrized tensor product of n single-particle Hilbert spaces H. If the identical particles are fermions, the n-particle states are vectors in an antisymmetrized tensor product of n single-particle Hilbert spaces H (see symmetric algebra and exterior algebra respectively). A general state in Fock space is a linear combination of n-particle states, one for each n.

Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H,

Here is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic or fermionic statistics, and the overline represents the completion of the space. The bosonic (resp. fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors (resp. alternating tensors ). For every basis for H there is a natural basis of the Fock space, the Fock states.

Definition edit

The Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space  


Here  , the complex scalars, consists of the states corresponding to no particles,   the states of one particle,   the states of two identical particles etc.

A general state in   is given by

  •   is a vector of length 1 called the vacuum state and   is a complex coefficient,
  •   is a state in the single particle Hilbert space and   is a complex coefficient,
  •  , and   is a complex coefficient, etc.

The convergence of this infinite sum is important if   is to be a Hilbert space. Technically we require   to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples   such that the norm, defined by the inner product is finite

where the   particle norm is defined by
i.e., the restriction of the norm on the tensor product  

For two general states

the inner product on   is then defined as
where we use the inner products on each of the  -particle Hilbert spaces. Note that, in particular the   particle subspaces are orthogonal for different  .

Product states, indistinguishable particles, and a useful basis for Fock space edit

A product state of the Fock space is a state of the form


which describes a collection of   particles, one of which has quantum state  , another   and so on up to the  th particle, where each   is any state from the single particle Hilbert space  . Here juxtaposition (writing the single particle kets side by side, without the  ) is symmetric (resp. antisymmetric) multiplication in the symmetric (antisymmetric) tensor algebra. The general state in a Fock space is a linear combination of product states. A state that cannot be written as a convex sum of product states is called an entangled state.

When we speak of one particle in state  , we must bear in mind that in quantum mechanics identical particles are indistinguishable. In the same Fock space, all particles are identical. (To describe many species of particles, we take the tensor product of as many different Fock spaces as there are species of particles under consideration). It is one of the most powerful features of this formalism that states are implicitly properly symmetrized. For instance, if the above state   is fermionic, it will be 0 if two (or more) of the   are equal because the antisymmetric (exterior) product  . This is a mathematical formulation of the Pauli exclusion principle that no two (or more) fermions can be in the same quantum state. In fact, whenever the terms in a formal product are linearly dependent; the product will be zero for antisymmetric tensors. Also, the product of orthonormal states is properly orthonormal by construction (although possibly 0 in the Fermi case when two states are equal).

A useful and convenient basis for a Fock space is the occupancy number basis. Given a basis   of  , we can denote the state with   particles in state  ,   particles in state  , ...,   particles in state  , and no particles in the remaining states, by defining


where each   takes the value 0 or 1 for fermionic particles and 0, 1, 2, ... for bosonic particles. Note that trailing zeroes may be dropped without changing the state. Such a state is called a Fock state. When the   are understood as the steady states of a free field, the Fock states describe an assembly of non-interacting particles in definite numbers. The most general Fock state is a linear superposition of pure states.

Two operators of great importance are the creation and annihilation operators, which upon acting on a Fock state add or respectively remove a particle in the ascribed quantum state. They are denoted   for creation and  for annihilation respectively. To create ("add") a particle, the quantum state   is symmetric or exterior- multiplied with  ; and respectively to annihilate ("remove") a particle, an (even or odd) interior product is taken with  , which is the adjoint of  . It is often convenient to work with states of the basis of   so that these operators remove and add exactly one particle in the given basis state. These operators also serve as generators for more general operators acting on the Fock space, for instance the number operator giving the number of particles in a specific state   is  .

Wave function interpretation edit

Often the one particle space   is given as  , the space of square-integrable functions on a space   with measure   (strictly speaking, the equivalence classes of square integrable functions where functions are equivalent if they differ on a set of measure zero). The typical example is the free particle with   the space of square integrable functions on three-dimensional space. The Fock spaces then have a natural interpretation as symmetric or anti-symmetric square integrable functions as follows.

Let   and  ,  ,  , etc. Consider the space of tuples of points which is the disjoint union


It has a natural measure   such that   and the restriction of   to   is  . The even Fock space   can then be identified with the space of symmetric functions in   whereas the odd Fock space   can be identified with the space of anti-symmetric functions. The identification follows directly from the isometric mapping


Given wave functions  , the Slater determinant

is an antisymmetric function on  . It can thus be naturally interpreted as an element of the  -particle sector of the odd Fock space. The normalization is chosen such that   if the functions   are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the permanent which gives elements of  -sector of the even Fock space.

Relation to the Segal–Bargmann space edit

Define the Segal–Bargmann space  [3] of complex holomorphic functions square-integrable with respect to a Gaussian measure:

Then defining a space   as the nested union of the spaces   over the integers  , Segal[4] and Bargmann showed[5][6] that   is isomorphic to a bosonic Fock space. The monomial
corresponds to the Fock state

See also edit

References edit

  1. ^ Fock, V. (1932). "Konfigurationsraum und zweite Quantelung". Zeitschrift für Physik (in German). 75 (9–10). Springer Science and Business Media LLC: 622–647. Bibcode:1932ZPhy...75..622F. doi:10.1007/bf01344458. ISSN 1434-6001. S2CID 186238995.
  2. ^ M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.
  3. ^ Bargmann, V. (1961). "On a Hilbert space of analytic functions and associated integral transform I". Communications on Pure and Applied Mathematics. 14: 187–214. doi:10.1002/cpa.3160140303. hdl:10338.dmlcz/143587.
  4. ^ Segal, I. E. (1963). "Mathematical problems of relativistic physics". Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II. Chap. VI.
  5. ^ Bargmann, V (1962). "Remarks on a Hilbert space of analytic functions". Proc. Natl. Acad. Sci. 48 (2): 199–204. Bibcode:1962PNAS...48..199B. doi:10.1073/pnas.48.2.199. PMC 220756. PMID 16590920.
  6. ^ Stochel, Jerzy B. (1997). "Representation of generalized annihilation and creation operators in Fock space" (PDF). Universitatis Iagellonicae Acta Mathematica. 34: 135–148. Retrieved 13 December 2012.

External links edit

  • Feynman diagrams and Wick products associated with q-Fock space - noncommutative analysis, Edward G. Effros and Mihai Popa, Department of Mathematics, UCLA
  • R. Geroch, Mathematical Physics, Chicago University Press, Chapter 21.