Functional Integral Representation of the Pauli-Fierz Model with Spin 1/2

0.54 MB·English

by  Fumio Hiroshima

Functional Integral Representation of
the Pauli-Fierz Model with Spin 1 2
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2 Fumio Hiroshima

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Department of Mathematics, University of Kyushu
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J 6-10-1, Hakozaki, Fukuoka, 812-8581, Japan

6 [email protected]
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] Jo´zsef Lo˝rinczi
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Zentrum Mathematik, Technische Universita¨t Mu¨nchen
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Boltzmannstr. 3, 85747 Garching bei Mu¨nchen, Germany
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a [email protected]
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School of Mathematics, Loughborough University
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v Loughborough LE11 3TU, United Kingdom
3 [email protected]
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X Abstract
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a
A Feynman-Kac-type formula for a L´evy and an infinite dimensional Gaussian
random process associated with a quantized radiation field is derived. In partic-
ular, a functional integral representation of e−tHPF generated by the Pauli-Fierz
Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics is con-
structed. When no external potential is applied H turns translation invariant
PF
⊕
and it is decomposed as a direct integral H = H (P)dP. The functional
PF R3 PF
integral representation of e−tHPF(P) is also given. Although all these Hamilto-
R
nians include spin, nevertheless the kernels obtained for the path measures are
scalarratherthanmatrixexpressions. Asanapplicationofthefunctionalintegral
representations energy comparison inequalities are derived.
1
2 The Pauli-Fierz model with spin
1 Introduction
Functional integration proved to be a useful approach in various applications to quan-
tum field theory. For the case of a quantum particle linearly coupled to a scalar boson
field, the so called Nelson model, it gives a tool to proving existence or absence of a
groundstate in Fockspace [Spo98,LMS02a]. Furthermore, groundstate properties can
bederivedintermsofpathmeasureexpectations[BHLMS02],andthequestionhowthe
model Hamiltonian and its ground state behave under lifting the so called infrared and
ultraviolet cutoffs can also be treated by the same method [LMS02b, GL07a, GL07b].
Another problem studied by this approach is that of the effective mass [BS05, Spo87].
Someoftheseresultshavebeenobtainedbyfunctionalintegrationonly, thussometimes
it offers a complementary method rather than a mere alternative.
In contrast with Nelson’s model, the Pauli-Fierz model describes a minimal cou-
pling of a particle to the quantized radiation field. The spectrum of the Pauli-Fierz
Hamiltonian has been extensively studied by a number of authors also using analytic
methods. In particular, the bottom of the spectrum of the Pauli-Fierz Hamiltonian
is contained in the absolutely continuous spectrum, no matter how small the coupling
constant is. Nevertheless, a ground state exists for arbitrary values of the coupling con-
stant without any infrared cutoff [BFS99, GLL01, LL03]. Functional integration is also
useful in studying the spectrum of the Pauli-Fierz Hamiltonian which was addressed
in the spinless case so far [BH07, Hir00a, Hir07, HL07].
The spinless Pauli-Fierz Hamiltonian is written as
1
Hˆ := ( i eA)2 +V +H (1.1)
PF rad
2 − ∇−
on L2(R3) L2(Q), where the former is the particle state space and the latter is
⊗
the state space of the quantum field, A stands for the vector potential, H for the
rad
photon field, and V is an external potential acting on the electron. These objects
will be explained in the following section in detail. The C -semigroup e−tHˆPF is defined
0
through spectral calculus. A functional integral representation of the semigroup e−tHˆPF
can be constructed on the space C([0, );R3) Q , involving a process consisting of
E
∞ ×
3-dimensional Brownian motion (B ) for the particle, and an infinite dimensional
t t≥0
Ornstein-UhlenbeckprocessonafunctionspaceQ forthefield[FFG97,Hab98,Hir97].
E
One immediate corollary for the functional integral representation is the diamagnetic
The Pauli-Fierz model with spin 3
inequality [AHS78, Hir97]
infσ( (1/2)∆+V +H ) infσ(Hˆ ). (1.2)
rad PF
− ≤
Using the fact that a path measure exists was also applied to proving self-adjointness
of Hˆ for arbitrary values of the coupling constant e [Hir00b, Hir02]. Furthermore,
PF
whenever Hˆ hasagroundstate, thepathmeasure canbeused toprove itsuniqueness
PF
[Hir00a] as an alternative to the methods making use of ergodic properties of the
semigroup in [Gro72, GJ68]. Other applications for the study of the ground state
include [BH07, HL07].
The pathmeasure of the coupled Brownian motionand Ornstein-Uhlenbeck process
can be written in terms of a mixture of two measures as the specific form of the
coupling between particle and field allows an explicit calculation of the Gaussian part.
The so obtained marginal over the particle is a Gibbs measure on Brownian paths
with densities dependent on the twice iterated Itˆo integral of a pair potential function
describing the effective field resulting from the Gaussian integration [Spo87, Hir00a,
BH07, GL07a].
Previous applications of rigorous functional integration to quantum field theory
covered, as far as we know, only cases when no spin was present in the model. In
this paper our main concern is to study by means of a Feynman-Kac-type formula the
Pauli-Fierz operator with spin 1/2. (1.1) is in this case replaced by
1
H := (~σ ( i eA))2 +V +H , (1.3)
PF rad
2 · − ∇−
where ~σ = (σ ,σ ,σ ) are the Pauli matrices standing for the spin (see details in the
1 2 3
next section). The random process of the particle modifies to a 3+1 dimensional joint
Wiener and jump process (ξ ) = (B ,σ ) , where the effect of the spin appears in
t t≥0 t t t≥0
the process σ = σ( 1)Nt hopping between the two possible values of the spin variable
t
−
σ, driven by a Poisson process (N ) . Our approach owes a debt to the ideas in
t t≥0
[ALS83], where a path integral representation of a C -semigroup generated by Pauli
0
operators in quantum mechanics was obtained by making use of an R3 Z -valued
2
×
process, with Z the additive group of order two. As we will see in the next subsection,
2
the Pauli operator is of a similar form as H , in fact both operators describe minimal
PF
interactions. While in [ALS83] only a path integral representation of operators with
non-vanishing off-diagonalelements was constructed, we improve onthis here since this
part of the spin interaction in general may have zeroes.
4 The Pauli-Fierz model with spin
Another model considered in the present paper is the so called translation invariant
Pauli-Fierz Hamiltonian which is the case of H above with zero external potential V.
PF
⊕
Translation invariance yields a fiber decomposition H = H (P)dP with respect
PF R3 PF
to total momentum Ptot, where the fiber Hamiltonian is given by
R
1
H (P) := (~σ (P P eA(0)))2 +H , P R3. (1.4)
PF f rad
2 · − − ∈
Here P denotes the momentum operator of the field. While the translation invariant
f
Hamiltonian does not have any point spectrum, H (P) under some conditions does
PF
[Fro74, Che01]. In [Hir07] the functional integral representation of e−tHˆPF(P) for the
spinless fiber Hamiltonian is constructed, where
1
Hˆ (P) := (P P eA(0))2 +H , P R3. (1.5)
PF f rad
2 − − ∈
Furthermore, uniqueness ofthegroundstateofHˆ (0)aswellastheenergycomparison
PF
inequality
ˆ ˆ
infσ(H (0)) infσ(H (P)) (1.6)
PF PF
≤
are shown.
Our mainpurpose inthis paper istoextend the results onthe spinless Hamiltonians
mentioned above to those with spin, i.e.,
(1) constructafunctionalintegralrepresentationofe−tHPF ande−tHPF(P) witha scalar
kernel;
(2) derive some energy comparison inequalities for H and H (P).
PF PF
We stress that H and H (P) include spin 1/2, nevertheless the kernels of their
PF PF
functional integrals obtained here are scalar. (1) is achieved in Theorems 4.11 and 5.2,
and (2) in Corollaries 4.13 and 5.4 below.
Here is an outline of the key steps of proving (1) and (2). First we assume that the
form factor ϕˆ is a sufficiently smooth function of compact support. Then we will see
that there exists a Pauli operator H0 (φ), φ Q, on L2(R3 Z ), which can be used
PF ∈ × 2
to define
⊕
H0 := H0 (φ)dµ(φ). (1.7)
PF PF
Q
Z
As it will turn out, for arbitrary values of the coupling constant e,
H = H0 +˙ H (1.8)
PF PF rad
The Pauli-Fierz model with spin 5
holds as an equality of self-adjoint operators (+˙ denotes quadratic form sum). Al-
though for weak couplings this results by the Kato-Rellich Theorem, it is non-trivial
for arbitrary values of e. Thus it will suffice to construct a functional integral rep-
resentation of the right hand side of (1.8). However, as was mentioned before, the
off-diagonal part of H0 (φ) may have in general zeroes or a compact support. In order
PF
to prevent the off-diagonal part vanish we change H0 (φ) for H0ε(φ) by adding a term
PF PF
controlled by a small parameter ε > 0. Then we work with
Hε := H0ε +˙ H (1.9)
PF PF rad
and obtain the original Hamiltonian by limε→0e−tHPεF = e−tHPF, where in fact
⊕
H0ε := H0ε(φ)dµ(φ).
PF PF
Q
Z
In particular, instead of for the semigroup e−tHPF, we construct the functional integral
representation of e−tHPεF. By the Trotter-Kato product formula we write
e−tHPεF = s lim (e−(t/n)HP0Fεe−(t/n)Hrad)n (1.10)
−n→∞
and derive the functional integral of the Pauli-operator e−tHP0Fε(φ) by using that the
form factor ϕˆ is chosen to be bounded and sufficiently smooth, with non-zero off-
diagonals. By making use of a hypercontractivity argument for second quantization
andtheMarkov propertyof projections, we areable toconstruct the functionalintegral
representation of e−tHPεF. An approximation argument on ϕˆ leads us then to our main
Theorem 4.11 for reasonable form factors.
The functional integral representation of e−tHPF(P) is further obtained by a combi-
nation of that of e−tHPF and [Hir07]. Since the functional integral kernels are scalar,
we can estimate (F,e−tHPFG) and (F,e−tHPF(P)G) directly, and derive some energy
| | | |
comparison inequalities.
Our paper is organized as follows. In Section 2 we discuss the Fock space re-
spectively Euclidean representations of the Pauli-Fierz Hamiltonian with spin 1/2 in
detail. Section 3 is devoted to discussing L´evy processes and functional integral rep-
resentations of Pauli operators. In Section 4 by using results of the previous section
and hypercontractivity properties of second quantization we construct the functional
integral representation of e−tHPF and derive comparison inequalities for ground state
energies. In Section 5 we derive the functional integral of e−tHPF(P) and obtain energy
6 The Pauli-Fierz model with spin
inequalities for this case. In Section 6 we comment on the multiplicity of ground states
of a model with spin. Section 7 is an appendix containing details on Poisson point
processes and a related Itˆo formula adapted to our context.
2 Function space representation of the Pauli-Fierz
model with spin
2.1 Pauli-Fierz model with spin 1/2 in Fock space
We begin by defining the Pauli-Fierz Hamiltonian as a self-adjoint operator.
Fock space Let := L2(R3 1,1 ) be the Hilbert space of a single photon, where
b
H ×{− }
R3 1,1 (k,j) are its momentum and polarization, respectively. Denote n-fold
×{− } ∋
symmetric tensor product by n , with 0 := C. The Fock space describing
sym symHb
the full photon field is defined then as the Hilbert space
N N
∞ n
F := (2.1)
b
H
" #
n=0 sym
M O
with scalar product
∞
(Ψ,Φ)F := (Ψ(n),Φ(n))⊗nsymHb, (2.2)
n=0
X
and Ψ = ∞ Ψ(n), Φ = ∞ Φ(n). Alternatively, F can be identified as the set
n=0 n=0
of ℓ -sequences Ψ(n) ∞ with Ψ(n) n . The vector Ω = 1,0,0,... F is
2 L { }n=0 L ∈ symHb { } ∈
called Fock vacuum. The finite particle subspace F is defined by
N fin
F := Ψ(n) ∞ F M N : Ψ(m) = 0, m M .
fin { }n=0 ∈ |∃ ∈ ∀ ≥
n o
Field operators With each f a photon creation and annihilation operator is
b
∈ H
associated. The creation operator a†(f) : F F is defined by
→
(a†(f)Ψ)(n) = √nS (f Ψ(n−1)), n 1,
n
⊗ ≥
where S (f f ) = (1/n!) f f is the symmetrizer with
n 1 ⊗ ··· ⊗ n π∈Πn π(1) ⊗ ··· ⊗ π(n)
respect to the permutation group Π of degree n. The domain of a†(f) is maximally
Pn
defined by
∞
D(a†(f)) := Ψ(n) ∞ n S (f Ψ(n−1)) 2 < .
{ }n=0 k n ⊗ k ∞
( (cid:12) )
(cid:12)Xn=1
(cid:12)
(cid:12)
(cid:12)
The Pauli-Fierz model with spin 7
The annihilation operator a(f) is introduced as the adjoint a(f) = (a†(f¯))∗ of a†(f¯)
with respect to scalar product (2.2). a†(f) and a(f) are closable operators, their closed
extensions will be denoted by the same symbols. Also, they leave F invariant and
fin
obey the canonical commutation relations on F :
fin
[a(f),a†(g)] = (f¯,g)1, [a(f),a(g)] = 0, [a†(f),a†(g)] = 0.
Second quantization and free field Hamiltonian Although the free field Hamiltonian
HF = k a†(k,j)a(k,j)dk
rad | |
j=±1Z
X
is usually given in terms of formal kernels of creation and annihilation operators, we
define it as the infinitesimal generator of a one-parameter unitary group since this
definition has advantages in studying functional integral representations. We use the
label F for objects defined in Fock space. This unitary group is constructed through a
functor Γ. Let C(X Y) denote the set of contraction operators from X to Y. Then
→
Γ : C( ) C(F F) is defined as
b b
H → H → →
∞
Γ(T) := [ nT]
⊗
n=0
M
for T C( ), where the tensor product for n = 0 is the identity operator. For
b b
∈ H → H
a self-adjoint operator h on , Γ(eith), t R, is a strongly continuous one-parameter
b
H ∈
unitary group on F. Then by Stone’s Theorem there exists a unique self-adjoint
operator dΓ(h) on F such that Γ(eith) = eitdΓ(h), t R. dΓ(h) is called the second
∈
quantization of h. The second quantization of the identity operator, N := dΓ(1) gives
thephotonnumber operator. Letω bethemultiplicationoperatorf ω (k)f(k,j) =
b b
7→
k f(k,j), k R3, j = 1 on . The operator HF := dΓ(ω ) is then the free field
| | ∈ ± Hb rad b
Hamiltonian.
Polarization vectors Two vectors e(k,+1) and e(k, 1), k = 0, are polarization
− 6
vectors whenever e(k, 1),e(k,+1),k/ k form a right-handed system in R3 with (1)
− | |
e(k, 1) e(k,+1) = k/ k , (2) e(k,j) e(k,j′) = δ , (3) e(k,j) k/ k = 0. We have
jj′
− × | | · · | |
k k
µ ν
e (k,j)e (k,j) = δ ,
µ ν µν − k 2
j=±1 | |
X
8 The Pauli-Fierz model with spin
independently of the specific choice of these vectors. One can choose the polarization
F
vectors atconvenience since theHamiltoniansH defined belowareunitaryequivalent
PF
up to this choice [Sas06].
Quantized radiationfield Notethata♯(f)islinearinf,wherea♯ = a,a†,thusformally
a♯(f) = f(k,j)a♯(k,j)dk. The quantized radiation field with ultraviolet cutoff
j=±1
function (form factor) ϕˆ is defined through the vector potentials
P R
1 ϕˆ(k) ϕˆ( k)
A (x) := e (k,j) a†(k,j)e−ik·x + − a(k,j)eik·x dk.
µ µ
√2 j=±1Z ωb(k) ωb(k) !
X
p p
Here ϕˆ is Fourier transform of ϕ. A standing assumption in this paper is
Assumption 2.1 We take ϕˆ(k) = ϕˆ( k) = ϕˆ(k) and √ω ϕˆ, ϕˆ/ω L2(R3).
b b
− ∈
Under Assumption 2.1A (x) isawell-defined symmetric operatorinF. Byk e(k,j) =
µ
·
0, the Coulomb gauge condition
3
[∂ ,A (x)] = 0,
xµ µ
µ=1
X
holds on F . By the fact that ∞ A (x)nΦ /n! < for Φ F , and Nelson’s
fin n=0k µ k ∞ ∈ fin
analytic vector theorem [RS75, TPh.X.39] it follows that Aµ(x)⌈Ffin is essentially self-
adjoint. We denote its closure Aµ(x)⌈Ffin by the same symbol Aµ(x).
Electron state space and Schr¨odinger Hamiltonian The Hilbert space describing the
electron is L2(R3;C2). Let σ ,σ ,σ be the 2 2 Pauli matrices
1 2 3
×
0 1 0 i 1 0
σ := , σ := − , σ := .
1 1 0 2 i 0 3 0 1
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) − (cid:21)
We have σ σ = δ + i 3 ǫαβγσ , where ǫαβγ is the totally antisymmetric tensor
α β αβ γ=1 γ
with ǫ123 = 1. Then the electron Hamiltonian on L2(R3;C2) with external potential V
P
is given by
3
1
H = (σ ( i ))2 +V. (2.3)
p µ µ
2 − ∇
µ=1
X
Here V acts as a multiplication operator and in some statements below it will be
required to satisfy one or both of the following conditions:
Assumption 2.2 Let V be
The Pauli-Fierz model with spin 9
(1) relatively bounded with respect to ( 1/2)∆ with a bound strictly less than 1;
−
(2) supx∈R3Ex e−2R0tV(Bs)ds < ∞, for all t ∈ (0,∞).
h i
(1) above is a usual ingredient for self-adjointness of Schr¨odinger operators. In (2)
the expectation Ex is meant under Wiener measure for 3-dimensional Brownian motion
(B ) startingatx. ItisinparticularsatisfiedbyKato-classpotentialswhichincludes
s s≥0
Coulomb potential.
Pauli-Fierz Hamiltonian The state space of the joint electron-field system is
F = L2(R3;C2) F. (2.4)
H ⊗
F
The non-interacting system is described by the totalfree Hamiltonian H 1+1 H .
p⊗ ⊗ rad
To define the quantized radiation field A we identify F with the set of C2 F-
H ⊗
valued L2 functions on R3, i.e., F = ⊕(C2 F)dx. Then we have by definition
H ∼ R3 ⊗
⊕
A = (1 A (x))dx. Hence (A F)(x) = A (x)F(x) for F(x) D(A (x)) and A
µ R3 ⊗ µ µ R µ ∈ µ µ
is self-adjoint. Taking into account the minimal interaction i i eA , we
R µ µ µ
− ∇ 7→ − ∇ −
obtain the Pauli-Fierz Hamiltonian
2
3
1
F F
H := σ ( i 1 eA ) +V 1+1 H (2.5)
PF 2 µ − ∇µ ⊗ − µ ⊗ ⊗ rad
!
µ=1
X
with coupling constant e R, i.e.,
∈
3
1 e
HF = ( i eA)2 +V +HF σ B , (2.6)
PF 2 − ∇− rad − 2 µ µ
µ=1
X
where we omit the tensor product for convenience and write
i ϕˆ(k)
B (x) = (k e(k,j)) a†(k,j)e−ik·x a(k,j)eik·x dk.
µ µ
−√2 × ω (k) −
j=±1Z b
X (cid:0) (cid:1)
p
In fact, B (x) = ( A(x)) , however, we regard A and B as independent operators
µ µ
∇×
in this paper.
F
A first natural question is whether H is a self-adjoint operator.
PF
Proposition 2.3 Under Assumption 2.1 HF is self-adjoint on D( ∆) D(HF ) and
PF − ∩ rad
F
bounded from below. Moreover, it is essentially self-adjoint on any core of H +H .
p rad
10 The Pauli-Fierz model with spin
Proof: See [Hir00b, Hir02]. qed
AspecialcaseconsideredinthispaperisthetranslationinvariantPauli-FierzHamil-
tonian obtained under V = 0. Then
eitPµtotHF e−itPµtot = HF , t R, µ = 1,2,3,
PF PF ∈
where Ptot denotes the total electron-field momentum
Ptot := i 1+1 PF
µ − ∇µ ⊗ ⊗ fµ
F
and P = dΓ(k ) is the momentum of the field. By translation invariance the Hilbert
fµ µ
F F
space and the Hamiltonian H can both be decomposed with respect to the
H PF
spectrum ofPtot as ⊕ F(P)dP andHF := ⊕K(P)dP,withaself-adjoint operator
R3H PF R3
F F
K(P) labeled by P on (P). It is seen that K(P) and (P) are isomorphic with
R R
H H
a self-adjoint operator resp. a Hilbert space. Define thus on C2 F the Pauli-Fierz
⊗
operator at total momentum P R3 by
∈
3
1 e
HF (P) := (P PF eA(0))2 +HF σ B (0). (2.7)
PF 2 − f − rad − 2 µ µ
µ=1
X
Then we have
Proposition 2.4 Under Assumption 2.1 HF (P), P R3, is self-adjoint on the do-
PF ∈
main D(HF ) 3 D((PF)2), and essentially self-adjoint on any core of the self-
rad µ=1 fµ
adjoint operator 1 3 (PF)2 + HF . Moreover, F = ⊕C2 FdP and HF =
T 2 µ=1 fµ rad H ∼ R3 ⊗ PF ∼
⊕ F
H (P)dP hold.
R3 PF P R
PR roof: See [Hir06, LMS06]. qed
Here is an incomplete list of results on the spectral properties of the Pauli-Fierz
Hamiltonian. TheexistenceofthegroundstateofH isestablishedin[BFS99,GLL01,
PF
LL03] and that of H (P) in [Fro74, Che01, HaHe06]. The multiplicity of the ground
PF
state is estimated in [Hir00a, HS01, BFP05, Hir06], a spectral scattering theory and
relaxation to ground states are studied in [Ara83a, Spo97, FGS01]. The perturbation
of embedded eigenvalues is reduced to investigating resonances [BFS98a, BFS98b].
Energyestimates areobtainedin[Fef96,FFG97,LL00]andtheeffective massisstudied
in [Spo87, CH04, HS05, Che06, BCFS06, HI07]. Related works on particle systems
interacting with quantum fields include [Ger00, BDG04, AGG04, LMS06, Sas06].