the use of fock spaces in quantum...

The Use of Fock Spaces in Quantum Mechanics

Maury LeBlanc

Department of MathematicsThe University of Georgia

29 June 2012

Motivation

A Fock space is an infinite-dimensional vector space and is anatural tool for quantum field theory. This mathematicalconstruction is used to construct the quantum states of amulti-particle system from a single particle system. The creationand annihilation operators are used to account for the introductionand removal of particles, allowing us to describe a system with avariable number of particles.

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Building Blocks of a Fock Space

Let H be a Hilbert space.

The typical example, with which I will be working in this talk, isL2(R4, d4x) with inner product:

〈φ|ψ〉 :=

φ(~x)†ψ(~x) d~x

where † denotes conjugate transpose, which in the case of C is justcomplex conjugation.

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DefinitionA tensor can best be thought of as a multi-dimensional array thatdescribes linear relationships between vectors.

I Scalars are a single number

I Vectors are an n-tuple of scalars with respect to a basis

I Tensors are one step farther in that they are systems of vectors(an m-tuple of vectors, i.e. an m-tuple of n-tuples of scalars.)

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For any positive integer m, define the m-fold tensor product

H⊗̂m = H⊗ · · · ⊗ H

This is the Hilbert space completion of finite linear combinations ofelements of the form v1 ⊗ · · · ⊗ vm with inner product

〈u1 ⊗ · · · ⊗ um, v1 ⊗ · · · ⊗ vm〉 = 〈u1, v1〉 · · · 〈um, vm〉

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H corresponds to a single-particle system.

The tensor product H⊗H describes a system consisting of twoidentical non-interacting particles. 1

H⊗̂m = H⊗ · · · ⊗ H describes the m-particle state.

C is used to describe the vacuum state.

1In quantum mechanics identical particles are indistinguishable. Thus in aFock space, all particles must be identical. In order to consider multiple typesof particles, one must take the tensor product of different Fock spaces - one foreach species.

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How do we put it all together?

The elements of the Cartesian product ×∞m=0H⊗m areinfinite-tuples of tensors:

{(T0,T1, . . . ,Tm, . . . )}

with Tm ∈ H⊗̂m

Consider the subset W ⊂ ×∞m=0H⊗m consisting of the elements forwhich all but finitely many of the Tm vanish.

W is an inner product space.

Define∞⊕

H⊗̂m to be the Hilbert space completion of W .

This space can then be used to characterize a system in whichthere is a variable number of particles.

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{(T0,T1, . . . ,Tm, . . . )}

with Tm ∈ H⊗̂m

Define∞⊕

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{(T0,T1, . . . ,Tm, . . . )}

with Tm ∈ H⊗̂m

Define∞⊕

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Two Quick Physics Definitions

There are two fundamental classes of subatomic particles:

I Bosons such as photons and gluons

I Fermions such as leptons (electrons and their relatives) andquarks (the building blocks of protons and neutrons)

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Formal Definition of a Fock Space

DefinitionA Fock space for bosons is the Hilbert space completion of thedirect sum of the symmetric tensors in the tensor powers of asingle-particle Hilbert space; while a Fock space for fermions usesanti-symmetric tensors. For the sake of simplicity, in this talk I willfocus on the bosonic Fock space.

F =∞⊕

SymH⊗̂m =

C⊕H⊕ (Sym (H⊗H))⊕ (Sym (H⊗H⊗H))⊕ · · ·

where Sym is the operator which symmetrizes a tensor.

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Formal Definition of a Fock Space

DefinitionA Fock space for bosons is the Hilbert space completion of thedirect sum of the symmetric tensors in the tensor powers of asingle-particle Hilbert space; while a Fock space for fermions usesanti-symmetric tensors. For the sake of simplicity, in this talk I willfocus on the bosonic Fock space.

F =∞⊕

SymH⊗̂m =

C⊕H⊕ (Sym (H⊗H))⊕ (Sym (H⊗H⊗H))⊕ · · ·

where Sym is the operator which symmetrizes a tensor.

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Symmetrizes a Tensor?! What does that even mean?

DefinitionA symmetric tensor is one that is invariant under any permutationof its vector indices.

That is to say that Tv1,v2...vm = Tvσ(1),vσ(2)...vσ(m)for any σ ∈ Sm

The symmetric part of a general tensor T of order m can be foundby computing:

SymT = 1m!

∑σ∈Sm

where τσTv1,v2...vm = Tvσ(1),vσ(2)...vσ(m)

A simple example being: Sym(φ⊗ ψ) = 12(φ⊗ ψ + ψ ⊗ φ)

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SymT = 1m!

∑σ∈Sm

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SymT = 1m!

∑σ∈Sm

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A typical element of FThe elements of a Fock space are infinite-tuples of physical fields.Physicists often denote their vectors using what is called ketnotation. Since I am working in their field, I have adapted thatnotation here.In ket notation, a general element of F is of the form:

|ψ〉 = |ψ0〉 ⊕ |ψ1〉 ⊕ |ψ2〉 ⊕ . . .

= a0|0〉 ⊕ a1|ψ1〉 ⊕∑i ,j

aij |ψ2i , ψ2j〉 ⊕ . . .

anda0, a1 ∈ C and aij = aji ∈ C

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A typical element of FThe elements of a Fock space are infinite-tuples of physical fields.Physicists often denote their vectors using what is called ketnotation. Since I am working in their field, I have adapted thatnotation here.In ket notation, a general element of F is of the form:

|ψ〉 = |ψ0〉 ⊕ |ψ1〉 ⊕ |ψ2〉 ⊕ . . .

= a0|0〉 ⊕ a1|ψ1〉 ⊕∑i ,j

aij |ψ2i , ψ2j〉 ⊕ . . .

anda0, a1 ∈ C and aij = aji ∈ C

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The inner product on FThe Fock space F is an infinite-dimensional complex Hilbert spacewith inner product

〈φ|ψ〉 :=∞∑

〈φm|ψm〉m

I The inner products on C and H are multiplication and theHilbert space inner product respectively.

I The inner product on Sym(H⊗H) is given by:

〈φ1 ⊗ ψ1|φ2 ⊗ ψ2〉 = 〈φ1|φ2〉〈ψ1|ψ2〉Explicitly this is the integral:∫

(φ1(~x1)ψ1(~x2))†φ2(~x1)ψ2(~x2) d4 ~x1d4 ~x2 =∫

φ1(~x1)†φ2(~x1) d4 ~x1

∫ψ1(~x2)†ψ2(~x2) d4 ~x2

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〈φ|ψ〉 :=∞∑

〈φm|ψm〉m

(φ1(~x1)ψ1(~x2))†φ2(~x1)ψ2(~x2) d4 ~x1d4 ~x2 =∫

φ1(~x1)†φ2(~x1) d4 ~x1

∫ψ1(~x2)†ψ2(~x2) d4 ~x2

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〈φ|ψ〉 :=∞∑

〈φm|ψm〉m

(φ1(~x1)ψ1(~x2))†φ2(~x1)ψ2(~x2) d4 ~x1d4 ~x2 =∫

φ1(~x1)†φ2(~x1) d4 ~x1

∫ψ1(~x2)†ψ2(~x2) d4 ~x2

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A typical element of F

The bosonic Fock space consists of all infinite sequences

(ψ0, ψ1, ψ2, . . .) such that∞∑

||ψm||2 =∞∑

〈ψm|ψm〉m <∞

The function ψn is an n-particle function

I ψ0 is a complex number

I ψ1 : R4 → C is an element of L2(R4) and of the formψ1 = ψ(~x) with ~x ∈ R4

I ψm : R4n → C is an element of L2sym(R4m) and is of the form

ψm = ψn(~x1, ~x2, . . . , ~xm) and is symmetric with respect to them arguments 2, each of which live in R4.

2This symmetry reflects the indistinguishableness for the m bosons12 / 15

Operators

The two important operators on the Fock space are the creationand annihilation operators which act by adding and removing aparticle, respectively, in the given quantum state.Fix u ∈ H

I The creation operator cm(u) : H⊗m → H⊗(m+1) is given by:

v1 ⊗ · · · ⊗ vm 7→√

m + 1 Sym (u ⊗ v1 ⊗ · · · ⊗ vm)

I The annihilation operator am(u) : H⊗m → H⊗(m−1) is givenby:

v1 ⊗ · · · ⊗ vm 7→√

m〈u, v1〉v2 ⊗ · · · ⊗ vm

The creation operator cm is also denoted a†m+1 since it is theadjoint to the annihilation operator am+1.These operators act as a basis for operators on the Fock space.

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Operators

v1 ⊗ · · · ⊗ vm 7→√

m + 1 Sym (u ⊗ v1 ⊗ · · · ⊗ vm)

v1 ⊗ · · · ⊗ vm 7→√

m〈u, v1〉v2 ⊗ · · · ⊗ vm

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Operators

v1 ⊗ · · · ⊗ vm 7→√

m + 1 Sym (u ⊗ v1 ⊗ · · · ⊗ vm)

v1 ⊗ · · · ⊗ vm 7→√

m〈u, v1〉v2 ⊗ · · · ⊗ vm

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Operators

v1 ⊗ · · · ⊗ vm 7→√

m + 1 Sym (u ⊗ v1 ⊗ · · · ⊗ vm)

v1 ⊗ · · · ⊗ vm 7→√

m〈u, v1〉v2 ⊗ · · · ⊗ vm

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Everything in its Right Place

We have therefore constructed a space on which we can describe anon-interactive system of quantum states and two operators whichallow us to account for a variable number of (bosonic) particles.

We can now begin to think about the simplest quantum mechanics.

But that is a story for another time.

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Thank you

Thank you for listening.

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References

I Stephane Attal “The Algebra of Canonical CommutationRelations” from his Expository Papers on his website:http://math.univ-lyon1.fr/∼attal/Mescours/fock.pdf

I Scott Glasgow “Creation and Annihilation Operators”from his Quantum Field Theory lecture notes on his website:

https://math.byu.edu/∼sag/QuantumFieldTheory/10-Creation%20and%20Annihilation%20Operators.pdf

I Michael Reed & Barry Simon Methods of MathematicalPhysics, Vol. II. Academic Press: New York, New York , 1975.

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Inner Product on W

W ⊂ ×∞m=0H⊗m consisting of the elements for which all butfinitely many of the Tm vanish.

The inner product on W is given by:〈(T0,T1, . . . ,Tm), (S0, S1, . . . ,Sn)〉 :=

〈T0,S0〉〈T1, S1〉 . . . 〈Tn, Sn〉

where m ≥ n and Tj = 0 for all j > m and Sk = 0 for all k > n

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Creation and Annihilation Operators are Adjoint - Example

Let v ∈ H, φ ∈ Sym(H⊗H), and ψ ∈ Sym(H⊗H⊗H)

〈a3(v)ψ, φ〉 = 〈√

3〈v , ψ1〉ψ2 ⊗ ψ3, φ1 ⊗ φ2〉 =√

3〈v , ψ1〉〈ψ2, φ1〉〈ψ3, φ2〉

whereas 〈ψ, c2(v)φ〉 = 〈ψ1 ⊗ ψ2 ⊗ ψ3,√

3Sym(v ⊗ φ1 ⊗ φ2)〉

= 〈ψ1 ⊗ ψ2 ⊗ ψ3,16 (v ⊗ φ1 ⊗ φ2 + v ⊗ φ2 ⊗ φ1+

φ1 ⊗ v ⊗ φ2 + φ1 ⊗ φ2 ⊗ v + φ2 ⊗ v ⊗ φ1 + φ2 ⊗ φ1 ⊗ v〉

36 (〈ψ1, v〉〈ψ2, φ1〉〈ψ3, φ2〉+ 〈ψ1, v〉〈ψ2, φ2〉〈ψ3, φ1〉+〈ψ1, φ1〉〈ψ2, v〉〈ψ3, φ2〉+ 〈ψ1, φ1〉〈ψ2, φ2〉〈ψ3, v〉+〈ψ1, φ2〉〈ψ2, v〉〈ψ3, φ1〉+ 〈ψ1, φ2〉〈ψ2, φ1〉〈ψ3, v〉)

These permutations of v ⊗ φ1 ⊗ φ2 while holding ψ constant is thesame as permuting ψ while holding v ⊗ φ1 ⊗ φ2 constant. Since ψis symmetric, the six terms in parentheses are equal. Thus

〈ψ, c2(v)φ〉 =√

3〈ψ1, v〉〈ψ2, φ1〉〈ψ3, φ2〉 = 〈a3(v)ψ, φ〉

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The annihilation operator does not require symmetrization

While the creation operator requires that we re-symmetrize after addingthe particle to the given state to ensure that we generate another bosonicstate, the annihilation operator produces a symmetric tensor.

Let∑

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

(i)m be a finite linear combination of states

which is symmetric (i.e. invariant under Sm).

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

ciam(v)(ψ

(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

ci 〈v , ψ(i)1 〉ψ

(i)2 ⊗ · · · ⊗ ψ

Fix an element σm−1 ∈ Sm−1 which permutes {2, . . . ,m}

and let it act on am(v)

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

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While the creation operator requires that we re-symmetrize after addingthe particle to the given state to ensure that we generate another bosonicstate, the annihilation operator produces a symmetric tensor.

Let∑

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

(i)m be a finite linear combination of states

which is symmetric (i.e. invariant under Sm).

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

ciam(v)(ψ

(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

ci 〈v , ψ(i)1 〉ψ

(i)2 ⊗ · · · ⊗ ψ

Fix an element σm−1 ∈ Sm−1 which permutes {2, . . . ,m}

and let it act on am(v)

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

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σm−1

ci 〈v , ψ(i)1 〉ψ

(i)2 ⊗ · · · ⊗ ψ

ci 〈v , ψ(i)1 〉ψ

(i)σ(2) ⊗ · · · ⊗ ψ

(i)σ(m)

= am(v)

ciψ(i)τ(1) ⊗ ψ

(i)τ(2) ⊗ · · · ⊗ ψ

(i)τ(m)

= am(v)

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

where τm = (1)σm−1 ∈ Sm

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The annihilation operator does not require symmetrizationSince the original tensor is symmetric,

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

Thus σm−1

ci 〈v , ψ(i)1 〉ψ

(i)2 ⊗ · · · ⊗ ψ

= am(v)

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

= am(v)

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

Hence am(v)

ciψ(i)1 ⊗ ψ

(i)2 ⊗ · · · ⊗ ψ

)is symmetric.

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Why is there a factor of√

The creation and annihilation operators have a factor of√

m whichseems superficially unimportant. The necessity of this factor ismade apparent in other quantum mechanical computations.

In quantum mechanics, the number operator is the observable thatcounts the number of particles in a state.It is usually defined as: Nm = cmam+1

A simple calculation (using a pure tensor) shows:Nm(v)(|ψ〉) = cm(v)am+1(v)(|ψ〉)

= cm(v)(√

m〈v , ψ1〉ψ2 ⊗ · · · ⊗ ψm

m Sym(v ⊗ (

√m〈v , ψ1〉ψ2 ⊗ · · · ⊗ ψm)

)= m〈v , ψ1〉Sym (v ⊗ ψ2 ⊗ · · · ⊗ ψm)

Hence the number operator Nm maps the m-particle space to itselfand also tells us how many particle are in the given quantum state.

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the use of fock spaces in quantum...

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