Download The Use of Fock Spaces in Quantum Mechanics

The Use of Fock Spaces in Quantum Mechanics
Maury LeBlanc
Department of Mathematics
The University of Georgia
29 June 2012
Motivation
A Fock space is an infinite-dimensional vector space and is a
natural tool for quantum field theory. This mathematical
construction is used to construct the quantum states of a
multi-particle system from a single particle system. The creation
and annihilation operators are used to account for the introduction
and removal of particles, allowing us to describe a system with a
variable number of particles.
1 / 15
Building Blocks of a Fock Space
Let H be a Hilbert space.
The typical example, with which I will be working in this talk, is
L2 (R4 , d 4 x) with inner product:
Z
hφ|ψi :=
φ(~x )† ψ(~x ) d~x
R4
where † denotes conjugate transpose, which in the case of C is just
complex conjugation.
2 / 15
Building Blocks of a Fock Space
Definition
A tensor can best be thought of as a multi-dimensional array that
describes 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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Building Blocks of a Fock Space
For any positive integer m, define the m-fold tensor product
d
m = H ⊗ ··· ⊗ H
H⊗
This is the Hilbert space completion of finite linear combinations of
elements of the form v1 ⊗ · · · ⊗ vm with inner product
hu1 ⊗ · · · ⊗ um , v1 ⊗ · · · ⊗ vm i = hu1 , v1 i · · · hum , vm i
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Building Blocks of a Fock Space
H corresponds to a single-particle system.
The tensor product H ⊗ H describes a system consisting of two
identical non-interacting particles. 1
d
m = H ⊗ · · · ⊗ H describes the m-particle state.
H⊗
C is used to describe the vacuum state.
1
In quantum mechanics identical particles are indistinguishable. Thus in a
Fock space, all particles must be identical. In order to consider multiple types
of particles, one must take the tensor product of different Fock spaces - one for
each species.
5 / 15
How do we put it all together?
⊗m
The elements of the Cartesian product ×∞
are
m=0 H
infinite-tuples of tensors:
{(T0 , T1 , . . . , Tm , . . . )}
d
m
with Tm ∈ H⊗
⊗m
Consider the subset W ⊂ ×∞
consisting of the elements for
m=0 H
which all but finitely many of the Tm vanish.
W is an inner product space.
Define
∞
M
H⊗m to be the Hilbert space completion of W .
d
m=0
This space can then be used to characterize a system in which
there is a variable number of particles.
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How do we put it all together?
⊗m
The elements of the Cartesian product ×∞
are
m=0 H
infinite-tuples of tensors:
{(T0 , T1 , . . . , Tm , . . . )}
d
m
with Tm ∈ H⊗
⊗m
Consider the subset W ⊂ ×∞
consisting of the elements for
m=0 H
which all but finitely many of the Tm vanish.
W is an inner product space.
Define
∞
M
H⊗m to be the Hilbert space completion of W .
d
m=0
This space can then be used to characterize a system in which
there is a variable number of particles.
6 / 15
How do we put it all together?
⊗m
The elements of the Cartesian product ×∞
are
m=0 H
infinite-tuples of tensors:
{(T0 , T1 , . . . , Tm , . . . )}
d
m
with Tm ∈ H⊗
⊗m
Consider the subset W ⊂ ×∞
consisting of the elements for
m=0 H
which all but finitely many of the Tm vanish.
W is an inner product space.
Define
∞
M
H⊗m to be the Hilbert space completion of W .
d
m=0
This space can then be used to characterize a system in which
there is a variable number of particles.
6 / 15
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) and
quarks (the building blocks of protons and neutrons)
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Formal Definition of a Fock Space
Definition
A Fock space for bosons is the Hilbert space completion of the
direct sum of the symmetric tensors in the tensor powers of a
single-particle Hilbert space; while a Fock space for fermions uses
anti-symmetric tensors. For the sake of simplicity, in this talk I will
focus on the bosonic Fock space.
F=
∞
M
SymH⊗m =
d
m=0
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
Definition
A Fock space for bosons is the Hilbert space completion of the
direct sum of the symmetric tensors in the tensor powers of a
single-particle Hilbert space; while a Fock space for fermions uses
anti-symmetric tensors. For the sake of simplicity, in this talk I will
focus on the bosonic Fock space.
F=
∞
M
SymH⊗m =
d
m=0
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?
Definition
A symmetric tensor is one that is invariant under any permutation
of 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 found
by computing:
X
1
τσ T
SymT = m!
σ∈Sm
where τσ Tv1 ,v2 ...vm = Tvσ(1) ,vσ(2) ...vσ(m)
A simple example being: Sym(φ ⊗ ψ) = 12 (φ ⊗ ψ + ψ ⊗ φ)
9 / 15
Symmetrizes a Tensor?! What does that even mean?
Definition
A symmetric tensor is one that is invariant under any permutation
of 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 found
by computing:
X
1
τσ T
SymT = m!
σ∈Sm
where τσ Tv1 ,v2 ...vm = Tvσ(1) ,vσ(2) ...vσ(m)
A simple example being: Sym(φ ⊗ ψ) = 12 (φ ⊗ ψ + ψ ⊗ φ)
9 / 15
Symmetrizes a Tensor?! What does that even mean?
Definition
A symmetric tensor is one that is invariant under any permutation
of 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 found
by computing:
X
1
τσ T
SymT = m!
σ∈Sm
where τσ Tv1 ,v2 ...vm = Tvσ(1) ,vσ(2) ...vσ(m)
A simple example being: Sym(φ ⊗ ψ) = 12 (φ ⊗ ψ + ψ ⊗ φ)
9 / 15
A typical element of F
The elements of a Fock space are infinite-tuples of physical fields.
Physicists often denote their vectors using what is called ket
notation. Since I am working in their field, I have adapted that
notation here.
In ket notation, a general element of F is of the form:
|ψi = |ψ0 i ⊕ |ψ1 i ⊕ |ψ2 i ⊕ . . .
= a0 |0i ⊕ a1 |ψ1 i ⊕
X
aij |ψ2i , ψ2j i ⊕ . . .
i,j
where |0i is a vector of length one and is called the vacuum state
|ψ1 i ∈ H is a state in the single particle Hilbert space
|ψ2i , ψ2j i = |ψ2i i ⊗ |ψ2j i ∈ Sym (H ⊗ H)
and
a0 , a1 ∈ C and aij = aji ∈ C
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A typical element of F
The elements of a Fock space are infinite-tuples of physical fields.
Physicists often denote their vectors using what is called ket
notation. Since I am working in their field, I have adapted that
notation here.
In ket notation, a general element of F is of the form:
|ψi = |ψ0 i ⊕ |ψ1 i ⊕ |ψ2 i ⊕ . . .
= a0 |0i ⊕ a1 |ψ1 i ⊕
X
aij |ψ2i , ψ2j i ⊕ . . .
i,j
where |0i is a vector of length one and is called the vacuum state
|ψ1 i ∈ H is a state in the single particle Hilbert space
|ψ2i , ψ2j i = |ψ2i i ⊗ |ψ2j i ∈ Sym (H ⊗ H)
and
a0 , a1 ∈ C and aij = aji ∈ C
10 / 15
The inner product on F
The Fock space F is an infinite-dimensional complex Hilbert space
with inner product
hφ|ψi :=
∞
X
hφm |ψm im
m=0
I
The inner products on C and H are multiplication and the
Hilbert space inner product respectively.
I
The inner product on Sym(H ⊗ H) is given by:
hφ1 ⊗ ψ1 |φ2 ⊗ ψ2 i = hφ1 |φ2 ihψ1 |ψ2 i
Explicitly this is the integral:
Z
(φ1 (x~1 )ψ1 (x~2 ))† φ2 (x~1 )ψ2 (x~2 ) d 4 x~1 d 4 x~2 =
Z
Z
†
4
φ1 (x~1 ) φ2 (x~1 ) d x~1
ψ1 (x~2 )† ψ2 (x~2 ) d 4 x~2
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The inner product on F
The Fock space F is an infinite-dimensional complex Hilbert space
with inner product
hφ|ψi :=
∞
X
hφm |ψm im
m=0
I
The inner products on C and H are multiplication and the
Hilbert space inner product respectively.
I
The inner product on Sym(H ⊗ H) is given by:
hφ1 ⊗ ψ1 |φ2 ⊗ ψ2 i = hφ1 |φ2 ihψ1 |ψ2 i
Explicitly this is the integral:
Z
(φ1 (x~1 )ψ1 (x~2 ))† φ2 (x~1 )ψ2 (x~2 ) d 4 x~1 d 4 x~2 =
Z
Z
†
4
φ1 (x~1 ) φ2 (x~1 ) d x~1
ψ1 (x~2 )† ψ2 (x~2 ) d 4 x~2
11 / 15
The inner product on F
The Fock space F is an infinite-dimensional complex Hilbert space
with inner product
hφ|ψi :=
∞
X
hφm |ψm im
m=0
I
The inner products on C and H are multiplication and the
Hilbert space inner product respectively.
I
The inner product on Sym(H ⊗ H) is given by:
hφ1 ⊗ ψ1 |φ2 ⊗ ψ2 i = hφ1 |φ2 ihψ1 |ψ2 i
Explicitly this is the integral:
Z
(φ1 (x~1 )ψ1 (x~2 ))† φ2 (x~1 )ψ2 (x~2 ) d 4 x~1 d 4 x~2 =
Z
Z
†
4
φ1 (x~1 ) φ2 (x~1 ) d x~1
ψ1 (x~2 )† ψ2 (x~2 ) d 4 x~2
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A typical element of F
The bosonic Fock space consists of all infinite sequences
∞
∞
X
X
2
(ψ0 , ψ1 , ψ2 , . . .) such that
||ψm || =
hψm |ψm im < ∞
m=0
m=0
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 form
ψ1 = ψ(~x ) with ~x ∈ R4
I
ψm : R4n → C is an element of L2sym (R4m ) and is of the form
ψm = ψn (x~1 , x~2 , . . . , x~m ) and is symmetric with respect to the
m arguments 2 , each of which live in R4 .
2
This symmetry reflects the indistinguishableness for the m bosons
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Operators
The two important operators on the Fock space are the creation
and annihilation operators which act by adding and removing a
particle, 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 given
by:
√
v1 ⊗ · · · ⊗ vm 7→ mhu, v1 iv2 ⊗ · · · ⊗ vm
†
The creation operator cm is also denoted am+1
since it is the
adjoint to the annihilation operator am+1 .
These operators act as a basis for operators on the Fock space.
13 / 15
Operators
The two important operators on the Fock space are the creation
and annihilation operators which act by adding and removing a
particle, 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 given
by:
√
v1 ⊗ · · · ⊗ vm 7→ mhu, v1 iv2 ⊗ · · · ⊗ vm
†
since it is the
The creation operator cm is also denoted am+1
adjoint to the annihilation operator am+1 .
These operators act as a basis for operators on the Fock space.
13 / 15
Operators
The two important operators on the Fock space are the creation
and annihilation operators which act by adding and removing a
particle, 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 given
by:
√
v1 ⊗ · · · ⊗ vm 7→ mhu, v1 iv2 ⊗ · · · ⊗ vm
†
since it is the
The creation operator cm is also denoted am+1
adjoint to the annihilation operator am+1 .
These operators act as a basis for operators on the Fock space.
13 / 15
Operators
The two important operators on the Fock space are the creation
and annihilation operators which act by adding and removing a
particle, 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 given
by:
√
v1 ⊗ · · · ⊗ vm 7→ mhu, v1 iv2 ⊗ · · · ⊗ vm
†
since it is the
The creation operator cm is also denoted am+1
adjoint to the annihilation operator am+1 .
These operators act as a basis for operators on the Fock space.
13 / 15
Everything in its Right Place
We have therefore constructed a space on which we can describe a
non-interactive system of quantum states and two operators which
allow 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.
14 / 15
Everything in its Right Place
We have therefore constructed a space on which we can describe a
non-interactive system of quantum states and two operators which
allow 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.
14 / 15
Everything in its Right Place
We have therefore constructed a space on which we can describe a
non-interactive system of quantum states and two operators which
allow 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.
14 / 15
Thank you
Thank you for listening.
15 / 15
References
I
Stephane Attal “The Algebra of Canonical Commutation
Relations” 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 Mathematical
Physics, Vol. II. Academic Press: New York, New York , 1975.
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Inner Product on W
⊗m
consisting of the elements for which all but
W ⊂ ×∞
m=0 H
finitely many of the Tm vanish.
The inner product on W is given by:
h(T0 , T1 , . . . , Tm ), (S0 , S1 , . . . , Sn )i :=
hT0 , S0 i hT1 , S1 i . . . hTn , Sn i
where m ≥ n and Tj = 0 for all j > m and Sk = 0 for all k > n
17 / 15
Creation and Annihilation Operators are Adjoint - Example
Let v ∈ H, φ ∈ Sym(H ⊗ H), and ψ ∈ Sym(H ⊗ H ⊗ H)
√
√
ha3 (v )ψ, φi = h 3hv , ψ1 iψ2 ⊗ ψ3 , φ1 ⊗ φ2 i = 3hv , ψ1 ihψ2 , φ1 ihψ3 , φ2 i
√
whereas hψ, c2 (v )φi = hψ1 ⊗ ψ2 ⊗ ψ3 , 3Sym(v ⊗ φ1 ⊗ φ2 )i
= hψ1 ⊗ ψ2 ⊗ ψ3 , 16 (v ⊗ φ1 ⊗ φ2 + v ⊗ φ2 ⊗ φ1 +
φ1 ⊗ v ⊗ φ2 + φ1 ⊗ φ2 ⊗ v + φ 2 ⊗ v ⊗ φ1 + φ2 ⊗ φ1 ⊗ v i
√
=
3
6
(hψ1 , v ihψ2 , φ1 ihψ3 , φ2 i + hψ1 , v ihψ2 , φ2 ihψ3 , φ1 i+
hψ1 , φ1 ihψ2 , v ihψ3 , φ2 i + hψ1 , φ1 ihψ2 , φ2 ihψ3 , v i+
hψ1 , φ2 ihψ2 , v ihψ3 , φ1 i + hψ1 , φ2 ihψ2 , φ1 ihψ3 , v i)
These permutations of v ⊗ φ1 ⊗ φ2 while holding ψ constant is the
same as permuting ψ while holding v ⊗ φ1 ⊗ φ2 constant. Since ψ
is symmetric, the six terms in parentheses are equal. Thus
√
hψ, c2 (v )φi = 3hψ1 , v ihψ2 , φ1 ihψ3 , φ2 i = ha3 (v )ψ, φi
18 / 15
The annihilation operator does not require symmetrization
While the creation operator requires that we re-symmetrize after adding
the particle to the given state to ensure that we generate another bosonic
state, the annihilation operator produces a symmetric tensor.
X
(i)
(i)
(i)
be a finite linear combination of states
Let
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
which is symmetric (i.e. invariant under Sm ).
!
X
(i)
(i)
(i)
am (v )
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm :=
i
X
X
(i)
(i)
(i)
(i)
(i)
(i)
=
ci hv , ψ1 iψ2 ⊗ · · · ⊗ ψm
ci am (v ) ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
i
Fix an element σm−1 ∈ Sm−1 which permutes {2, . . . ,!
m}
X
(i)
(i)
(i)
and let it act on am (v )
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
19 / 15
The annihilation operator does not require symmetrization
While the creation operator requires that we re-symmetrize after adding
the particle to the given state to ensure that we generate another bosonic
state, the annihilation operator produces a symmetric tensor.
X
(i)
(i)
(i)
be a finite linear combination of states
Let
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
which is symmetric (i.e. invariant under Sm ).
!
X
(i)
(i)
(i)
am (v )
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm :=
i
X
X
(i)
(i)
(i)
(i)
(i)
(i)
=
ci hv , ψ1 iψ2 ⊗ · · · ⊗ ψm
ci am (v ) ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
i
Fix an element σm−1 ∈ Sm−1 which permutes {2, . . . ,!
m}
X
(i)
(i)
(i)
and let it act on am (v )
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
19 / 15
The annihilation operator does not require symmetrization
!
σm−1
X
(i)
(i)
ci hv , ψ1 iψ2
⊗ ··· ⊗
(i)
ψm
=
X
i
i
= am (v )
(i)
X
= am (v ) τm
(i)
(i)
(i)
ci hv , ψ1 iψσ(2) ⊗ · · · ⊗ ψσ(m)
!
(i)
(i)
ci ψτ (1) ⊗ ψτ (2) ⊗ · · · ⊗ ψτ (m)
i
!!
X
(i)
(i)
(i)
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
where τm = (1)σm−1 ∈ Sm
20 / 15
The annihilation operator does not require symmetrization
Since the original tensor is symmetric,
!
X
(i)
(i)
(i)
τm
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
!
X
=
(i)
ci ψ1
⊗
(i)
ψ2
⊗ ··· ⊗
(i)
ψm
i
!
Thus σm−1
X
(i)
(i)
ci hv , ψ1 iψ2
⊗ ··· ⊗
(i)
ψm
i
!!
= am (v ) τm
X
(i)
(i)
(i)
(i)
ψ2
(i)
ψm
ci ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψm
i
!
= am (v )
X
(i)
ci ψ1
⊗
⊗ ··· ⊗
i
!
Hence am (v )
X
i
(i)
ci ψ1
⊗
(i)
ψ2
⊗ ··· ⊗
(i)
ψm
is symmetric.
21 / 15
Why is there a factor of
√
m?
√
The creation and annihilation operators have a factor of m which
seems superficially unimportant. The necessity of this factor is
made apparent in other quantum mechanical computations.
In quantum mechanics, the number operator is the observable that
counts the number of particles in a state.
It is usually defined as: Nm = cm am+1
A simple calculation (using a pure tensor) shows:
Nm (v )(|ψi) = cm (v )am+1 (v )(|ψi)
√
= cm (v ) mhv , ψ1 iψ2 ⊗ · · · ⊗ ψm
√
√
= m Sym v ⊗ ( mhv , ψ1 iψ2 ⊗ · · · ⊗ ψm )
= mhv , ψ1 iSym (v ⊗ ψ2 ⊗ · · · ⊗ ψm )
Hence the number operator Nm maps the m-particle space to itself
and also tells us how many particle are in the given quantum state.
22 / 15