Poincaré group

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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 paper Konfigurationsraum und zweite Quantelung.[1][2]

Informally, a Fock space consists of a set of Hilbert spaces representing a zero particle state, a one particle state, a two particle state, and so on. If the identical particles are bosons, the n-particle state is a symmetrized tensor product of n single particle Hilbert spaces H. If the identical particles are fermions, the n-particle state is an antisymmetrized tensor product of n single particle Hilbert spaces H. A state in Fock space is a linear combination of states, where each state has a definite number of particles.

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:

Fν(H)=n=0SνHn

Here Sν is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic (ν=+) or fermionic (ν=) 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 F+(H)=S*H (resp. alternating tensors F(H)=*H). For every basis for H there is a natural basis of the Fock space, the Fock states.

Definition

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

Fν(H)=n=0SνHn=H(Sν(HH))(Sν(HHH))

Here , a complex scalar, represents the states of no particles, H the state of one particle, Sν(HH) the states of two identical particles etc.

A typical state in Fν(H) is given by

|Ψν=|Ψ0ν|Ψ1ν|Ψ2ν=a0|0|ψ1ijaij|ψ2i,ψ2jν

where

|0 is a vector of length 1, called the vacuum state and a0 is a complex coefficient,
|ψ1H is a state in the single particle Hilbert space,
|ψ2iψ2jν=12(|ψ2i|ψ2j+(1)ν|ψ2j|ψ2i)Sν(HH), and aij=νaji is a complex coefficient
etc.

The convergence of this infinite sum is important if Fν(H) is to be a Hilbert space. Technically we require Fν(H) to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples |Ψν=(|Ψ0ν,|Ψ1ν,|Ψ2ν,) such that the norm, defined by the inner product is finite

|Ψνν2=n=1Ψn|Ψnν<

where the n particle norm is defined by

Ψn|Ψnν=limMi1,in,j1,jn<Mai1,,in*aj1,,jnψi1|ψj1ψin|ψjn

i.e. the restriction of the norm on the tensor product Hn

For two states

|Ψν=|Ψ0ν|Ψ1ν|Ψ2ν=a0|0|ψ1ijaij|ψ2i,ψ2jν, and
|Φν=|Φ0ν|Φ1ν|Φ2ν=b0|0|ϕ1ijbij|ϕ2i,ϕ2jν

the inner product on Fν(H) is then defined as

Ψ|Φν:=nΨn|Φnν=a0*b0+ψ1|ϕ1+ijklaij*bklϕ2i|ψ2kψ2j|ϕ2lν+

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

Example

An example of a pure state of the Fock space is

|Ψν=|ϕ1,ϕ2,,ϕnν=|ϕ1|ϕ2|ϕn

which describes a collection of n particles, one of which has quantum state ϕ1, another ϕ2 and so on up to the nth particle, where each ϕi is any state from the single particle Hilbert space H. Here juxtaposition is symmetric respectively antisymmetric multiplication in the symmetric and antisymmetric tensor algebra. The general state in a Fock space is a linear combination of pure states. A state that cannot be written as a product of pure states is called an entangled state.

When we speak of one particle in state ϕi, it must be borne 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, 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 |Ψ is fermionic, it will be 0 if two (or more) of the ϕi are equal because the anti symmetric (exterior) product |ϕi|ϕi=0. This is a mathematical formulation of the Pauli exclusion principle that no two (or more) fermions can be in the same quantum state. 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 the choice of a basis {|ψi}i=0,1,2, of H, we can denote the state with n0 particles in state |ψ0, n1 particles in state |ψ1, ..., nk particles in state |ψk by

|n0,n1,,nkν,=|ψ0n0|ψ1n1|ψknk

where each ni takes the value 0 or 1 for fermionic particles and 0, 1, 2, ... for bosonic particles. Such a state is called a Fock state. When the |ψi 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 pure state is the linear superposition of Fock states.

Two operators of great importance are the creation and annihilation operators, which upon acting on a Fock state add respectively remove a particle in the ascribed quantum state. They are denoted a(ϕ) and a(ϕ) respectively, with the quantum state |ϕ the particle which is "added" by multiplication with |ϕ respectively "removed" by (even or odd) interior product with ϕ| which is the adjoint of a(ϕ). It is often convenient to work with states of the basis of H so that these operators remove and add exactly one particle in the given basis state. These operators also serve as a basis for more general operators acting on the Fock space, for instance the number operator giving the number of particles in a specific state |ϕi is a(ϕi)a(ϕi).

Wave Function Interpretation

Often the one particle space H is given as L2(X,μ), the space of square-integrable functions on a space X with measure μ (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 H=L2(3,d3x) 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 X0={*} and X1=X, X2=X×X, X3=X×X×X etc. Consider the space of tuples of points which is the disjoint union

X*=X0X1X2X3.

It has a natural measure μ* such that μ*(X0)=1 and the restriction of μ* to Xn is μn. The even Fock space F+(L2(X,μ)) can then be identified with the space of symmetric functions in L2(X*,μ*) whereas odd Fock space F(L2(X,μ)) can be identified with the space of anti-symmetric functions. The identification follows directly from the isometric mapping

L2(X,μ)nL2(Xn,μn)
ψ1(x)ψn(x)ψ1(x1)ψn(xn).

Given wave functions ψ1=ψ1(x),,ψn=ψn(x), the Slater determinant

Ψ(x1,xn)=1n!|ψ1(x1)ψn(x1)ψ1(xn)ψn(xn)|

is an antisymmetric function on Xn. It can thus be naturally interpreted as an element of n-particle section of the odd Fock space. The normalisation is chosen such that Ψ=1 if the functions ψ1,,ψn are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the permanent which gives elements of n-sector of the even Fock space.

Relation to Bargmann-Fock space

Define a space Bn[3] of complex holomorphic functions convergent with respect to a Gaussian measure:

2(n)={f:nf2(n)<}, where f2(n):=n|f(z)|2eπ|z|2dz

Then defining a space B as the amalgamation of spaces Bn over the integers n1, Bargmann in 1961 showed [4] [5] that B is isomorphic to a bosonic Fock space.

See also

References

  1. V. Fock, Z. Phys. 75 (1932), 622-647
  2. M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.
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    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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