Mathematical Theory of Functional
Integrals for Bose Systems
Tadeusz Balaban
Rutgers University
Joel Feldman
University of British Columbia
Horst Knörrer, Eugene Trubowitz
ETH-Zürich
http://www.math.ubc.ca/e feldman/
http://www.math.ubc.ca/e feldman/andrejewski.html
F1
Abstract
Functional integrals have long been used, formally, to
provide intuition about the behaviour of quantum field
theories. For the past several decades, they have also
been used, rigorously, in the construction and analysis of
those theories. I will talk about the rigorous derivation
of some functional integral representations for the partition function and correlation functions of (cutoff) many
Boson systems that provide a suitable starting point for
their construction.
Outline
I
I
I
I
The Physical Setting
The Goal
One Result
The Main Ideas Leading to the Results
F2
The Physical Setting
Consider a gas of small (i.e. quantum mechanical effects
are important) particles that are bosons.
I Each particle has a kinetic energy. The corresponding
quantum mechanical observable is an operator h. The
1
most commonly used h = − 2m
∆.
I The particles interact with each other through a repulsive two-body potential, v(x, y).
I The system is in thermodynamic equilibrium through
contact with a heat bath. The gas and the heat bath
can exchange both energy and particles. The probability distributions for the energy and for the number of
particles in the gas are controlled by the temperature
1
T = kβ
and chemical potential µ respectively.
I Two interesting observables for this system are the
Hamiltonian H (built from h and v) and the number
operator N .
I I will concentrate on the partition function
Z = Tr e−β(H−µN )
F3
The Goal
Formally,
Tr e
−β(H−µN )
=
where
∗
D(φ , φ) =
Z
∗
D(φ , φ) e
A(φ∗ ,φ)
(1)
φβ =φ0
Y
dφ∗
τ (x) dφτ (x)
2πi
x∈IR3
0≤τ ≤β
and
A(φ∗ , φ) =
Z
β
dτ
0
Z
∂
φτ (x)
d3 x φ∗τ (x) ∂τ
IR3
−
Z
β
dτ K
0
φ∗τ , φτ
and
K(φ∗ , φ) =
ZZ
dxdy φ(x)∗ h(x, y)φ(y)
Z
− µ dx φ(x)∗ φ(x)
ZZ
+ 21
dxdy φ(x)∗ φ(y)∗ v(x, y) φ(x)φ(y)
F4
If φτ (x) = Φ ∈ C is a constant, independent of τ and
x, the exponent A(φ∗ , φ) simplifies to minus the integral
over τ and x of the “naive effective potential”
4
1
v̂(0)|Φ|
2
− µ|Φ|2
R
where v̂(0) = dy v(x, y), and we have assumed that
v(x, y) is translation invariant and that h annihilates constants.
The minimum of this effective potential is
B nondegenerate at the point Φ = 0 when µ < 0 and
q
µ
B degenerate along the circle |Φ| =
v̂(0) when µ > 0.
µ>0
µ<0
|Φ|
Both sides of (1) require careful definition.
F5
To carefully define the left hand side you take a limit
of obviously well–defined approximations. One way to
get (pretty) obviously well–defined approximation is to
replace space IR3 by a finite number of points X. Then
I The space of all states of this system is
F=
I
∞
L
n=0
Fn with Fn =
L2s (X n )
=C
|X|n
/Sn
Tr e−β(H−µN ) is well–defined because
H, N : Fn → Fn
N Fn = n1l
H Fn ≥ (Cn−D)n1l
dim Fn ≤ |X|n
=⇒ Tr Fn e−β(H−µN ) ≤ e−β(Cn
∞
X
=⇒
T rFn e−β(H−µN ) < ∞
2
−Dn−µn)
|X|n
n=0
F6
I
The analog of (1) is again
Tr e−β(H−µN ) =
Z
D(φ∗ , φ) eA(φ
∗
,φ)
(2)
φβ =φ0
but with
∗
D(φ , φ) =
and
R
IR3
Y
dφ∗
τ (x) dφτ (x)
2πi
x∈X
0≤τ ≤β
d3 x replaced by
Z
dx = (cell volume)
X
x∈X
The goal is to get a rigorous version of (2) which can
provide a useful starting point for a study of the limit
X → IR3 .
F7
WARNING
The exponent A(φ∗ , φ) is complex.
A(φ∗ ,φ)
=⇒ e
oscillates wildly
1
∗
part ofA(φ∗ ,φ)
cannot be turned into
=⇒ const D(φ , φ)e
an ordinary well–defined complex measure on some space
of paths, in contrast to Wiener measure.
For example, if
1
~ = R
dµn (φ)
~
~
~
e− 2 σφ·Cn φ dn φ
~
~
~
e− 2 σφ·Cn φ dn φ
IRn
1
with Cn > 0, Re σ > 0 and Im σ 6= 0, then
Z
R
~ nφ
~ n~
− 12 (Re σ)φ·C
d φ
e
n
~ = IRR
dµn (φ)
1 ~
~ n ~ σ
φ·C
φ
−
n
n
2
IR
d φ
e
IRn
Re σ −n/2 n→∞
−→ ∞
= |σ|
=⇒ The rigorous version is not going to be very aesthetically satisfying.
F8
One Result
Notation:
Tp =
εp =
dµp,r (φ∗ , φ) =
τ=
q βp
β
p
q = 1, ··· , p
Y Y h dφ∗ (x) dφ
τ
τ (x)
2πı
τ ∈Tp x∈X
χ(|φτ (x)| < r)
i
Theorem. Suppose that the sequence R(p) → ∞ as
p → ∞ at a suitable rate. Then
Tr e−β (H−µN )
Z
= lim dµp,R(p) (φ∗ , φ)
p→∞
Q −R dy [φ∗τ (y)−φ∗τ −ε (y)]φτ (y) −εp K(φ∗τ −ε ,φτ )
p
p
e
e
τ ∈Tp
with the convention that φ0 = φβ .
F9
The Main Ingredients – Coherent States
If |X| = 1, then
F=
∞
L
n=0
Fn with Fn = C
Let en = 1 ∈ C = Fn . For each φ ∈ C the coherent
state
∞
X
√1 φn en ∈ F
|φi =
n!
n=0
is an eigenvector for the field (or annihilation) operator.
ψen =
√
n en−1
The inner product between two coherent states is
α γ = eα γ
For general X, there is a similar coherent state | φ i for
each φ ∈ C|X| . The inner product between two coherent
states is
R
α γ = e dy α(y) γ(y)
F 10
The Main Ingredients – Resolution of the identity
Formally, the analog of
1lv =
X
(en , v) en
n
for coherent
states is
Z Y
R
i
h ∗
dφ (x) dφ(x)
−
1l =
e
2πı
dy |φ(y)|2
x∈X
Here
|φihφ|
| φ i h φ | : v ∈ F 7→ inner product of v and | φ i
Theorem. For each r > 0, let
R
i
Y hZ
−
dφ∗ (x) dφ(x)
e
Ir =
2πı
x∈X
dy |φ(y)|2
|φ(x)|<r
|φi
|φihφ|
(a) 0 < Ir < 1l.
(b) Ir commutes with N .
(c) Ir converges strongly to the identity operator as
r → ∞.
(d) For all n and r, the operator norm
(1l − Ir ) Fn ≤ |X| 2n+1 e−r2 /2
F 11
Proof: It is easy to guess an orthonormal basis of eigenvectors for Ir and to find all of the eigenvalues:
If |X| = 1, then
∞
L
F=
Fn with Fn = C
n=0
and em = 1 ∈ C = Fm m = 0, 1, 2, 3, · · ·
is an
orthonormal basis for F. Recall that
∞
X
√1 φn en
|φi =
n!
n=0
So
Ir e m =
=
=
=
Z
Z
dφ̄ dφ
2πı
|φ|<r
e
−|φ|2
| φ i φ em
dφ̄ dφ −|φ|2
√1 φ̄m
e
|
φ
i
2πı
m!
|φ|<r
Z
∞
X
dφ̄ dφ −|φ|2
√ 1√
e
2πı e
n! m! n
|φ|<r
n=0
1
m!
Z
= 1−
φ̄m φn
|φ|2m em
dφ̄ dφ −|φ|2
2πı e
|φ|<r
Z ∞
−t m
1
e
t dt em
m!
2
r
F 12
The Main Ingredients – Trace
Formally, the analog of
Tr B =
X
(en , Ben )
n
for coherent states is
Z Yh
R
i
∗
−
dφ (x) dφ(x)
Tr B =
e
2πı
x∈X
dy |φ(y)|2
hφ |B | φi
Proposition. Let B be a bounded operator on F. For
all r > 0, BIr is trace class and
Y hZ dφ∗ (x) dφ(x) i − R dy |φ(y)|2
e
hφ |B | φi
Tr BIr =
2πı
x∈X
|φ(x)|<r
If |X| = 1,
P
Tr BIr =
h em | BIr | em i
Z m
P
dφ̄ dφ −|φ|2
h e m | B | φ i φ em
=
2πı e
m |φ|<r
Z
dφ̄ dφ −|φ|2 P
=
φ em h e m | B | φ i
2πı e
m
|φ|<r
Z
dφ̄ dφ −|φ|2
hφ |B | φi
=
2πı e
Proof:
|φ|<r
F 13
Combining the previous two results,
Q − β (H−µN )
−β(H−µN )
Tr e
= Tr
e p
= lim Tr
p→∞
= lim
p→∞
Q
τ ∈Tp
e
−β
p (H−µN )
IR(p)
τ ∈Tp
Y hZ
x∈X
τ ∈Tp
dφ∗
τ (x) dφτ (x)
2πı
|φτ (x)|<R(p)
Q D
τ ∈Tp
φτ − β
p
e
−|φτ (x)|2
i
β
E
− p (H−µN ) φτ
e
Proposition. There are constants C, c such that the
following holds. For each ε > 0, there is an analytic
function F (ε, α∗ , φ) such that
E
D −ε(H−µN ) F (ε,α∗ ,φ)
α e
φ =e
on the domain kαk∞ , kφk∞ < C √1ε . Write
Z
F (ε, α∗, φ) = dx α(x)∗ φ(x) − εK(α∗, φ) + F0 (ε, α∗, φ)
X
Then
|F0 (ε, α∗ , φ)| ≤ c ε2 (Φ2 + kvk21,∞ Φ6 )
for all 0 < ε ≤ 1 and all kαk∞ , kφk∞ ≤ Φ ≤ 12 C √1ε .
F 14
−ε(H−µN ) φ is an entire funcIdea of Proof: α e
tion of α∗ and φ Rand a C ∞ function of ε for ε ≥ 0.
α∗ (x)φ(x) dx
Since α φ = e
6= 0, the matrix element
has the representation
E
D −ε(H−µN ) F (ε,α∗ ,φ)
α e
φ
=
e
in a neighbourhood of 0, with F (ε, α∗ , φ) is analytic in
α∗ , φ. F satisfies the differential equation
∂
∂ε F
= −K(α∗ , ∂∂α∗ )F
ZZ
F
∂ F
−
dxdy α(x)∗ α(y)∗ v(x, y) ∂∂α(x)
∗ ∂α(y)∗
X
with the initial condition
∗
F (0, α , φ) = ln α φ =
Z
dx α(x)∗ φ(x)
X
It is tedious but straight forward to convert this into a
system of integral equations for coefficients in the Taylor
expansion of F (ε, α∗ , φ) in powers of α∗ and φ. The
system can be solved and bounded by iteration.
F 15
So we now have
Tr e−β (H−µN )
Z
= lim
dµp,R(p) (φ∗ , φ)
p→∞
Q −R dy [φ∗τ (y)−φ∗τ −ε (y)]φτ (y) −εp K(φ∗τ −ε ,φτ )
p
p
e
e
τ ∈Tp
Q
e
−F0 (εp ,φ∗
τ −εp,φτ )
τ ∈Tp
and we just have to eliminate the F0 ’s. The sum
X
τ ∈Tp
F0 (εp , φ∗τ −εp, φτ )
I
has p terms.
I
Each term is bounded by
1
p2
times an unbounded function of the |φτ |’s.
F 16
Tr e−β(H−µN )
= lim Tr e
p→∞
= lim Tr e
p→∞
−β
p (H−µN )
1le
−β
p (H−µN )
−β
p (H−µN )
IR(p) e
Z ↓
d2 φ β
−β
p (H−µN )
IR(p) · · · e
Z ↓
d 2 φ2 β
p
= lim
p→∞
YhZ
x∈X
1≤j<p
1l · · · e
j
j
β
p
1l
−β
p (H−µN )
IR(p)
p
dφ∗ β (x) dφ
|φ
−β
p (H−µN )
p
j
β
p
2πı
(x)|<R(p)
(x)
e
−|φ
j
β
p
(x)|2 i
Tr
β
ED
ED
(H−µN
)
(H−µN
)
−β
−
φβ e p
φ2 β e p
φβ
φ2 β
p
p
p
p
β
D
· · · φ(p−1) β e− p (H−µN ) IR(p)
p
= lim
p→∞
Y hZ
x∈X
τ ∈Tp
= lim
p→∞
Z
dφ∗
τ (x) dφτ (x)
2πı
|φτ (x)|<R(p)
Q D
τ ∈Tp
φτ − β
dµp,R(p) (φ∗ , φ) exp
p
e−|φτ (x)|
2
i
β
E
− p (H−µN ) e
φτ
nX
τ ∈Tp
F (εp , φ∗τ − β , φτ )
p
o
F 17
References
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[FH] Richard P. Feynman and Albert R. Hibbs, Quantum
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[GJ] James Glimm and Arthur Jaffe, Quantum physics:
a functional integral point of view, Springer–
Verlag, 1987.
[NO] John W. Negele and Henri Orland, Quantum
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