Explicit gauge invariant quantization of the Schwinger model on a circle in the functional Schrödinger representation

21 Pages·0.2 MB·English

by  Joakim Hallin

G¨oteborg
ITP 95-29
hep-th/9601136
December 1995
6
9
9 Explicit gauge invariant quantization of the Schwinger model on
1
a circle in the functional Schr¨odinger representation
n
a
J Joakim Hallin1 and Per Liljenberg2

5
2

1
Institute of Theoretical Physics
v
6 Chalmers University of Technology
3 and University of G¨oteborg
1
1 S-412 96 G¨oteborg, Sweden
0
6 Abstract
9
/
h WesolvetheSchwinger modelonacirclebyfirstfindingtheexplicitgroundstate
t functional(s). Havingdonethis,wegivethestructureoftheHilbertspaceandderive
-
p
bosonization formulae in this formalism.
e
h
:
v
i
X
r
a
1
Email address: [email protected]
2
Email address: [email protected]
1 Introduction and Discussion
Massless QED in 1+1 dimensions was first studied by Schwinger [1]. Since then it has
been considered by numerous authors. In particular it has been studied on the circle by
[2, 3, 4].
The spectrum of the theory is equivalent to that of a massive free boson. From
a fermionic point of view this result is non-trivial. In this paper we are treating the
Schwinger model in a functional representation. This has some nice properties. We treat
gauge-invariancein Dirac’sway i.e. we let states beannihilated by theconstraint operator
and we only allow operators commuting with the constraint operator. In a functional
representation this can be done explicitly. Furthermore, the functional representation
space islargeenoughto containtheinequivalent Fockspaces thatalways exist inquantum
field theory. In the case of the Schwinger model this is desirable since the solution of
the model usually involves a Bogoliubov transformation that takes you out of a chosen
Fockspace into another. The functional representation allows you to treat all these Fock
spaces in a general setting.
One of the main motivations for this paper is the the study of the groundstate func-
tional for the Schwinger model, expressed in terms of fermionic variables. The idea is
that the groundstate in more complicated theories, like the massive Schwinger model or
higher dimensional QED, may be of the same or a similar form as that of the Schwinger
model. Even if this does not turn out to be the case, one might use it as a variational
ansatz for such theories.
So what does the groundstate look like? The groundstate cannot be gaussian in a
fermionic representation since the spectrum is not that of a free fermion. It turns out
that the groundstate is a two parameter functional where one of the parameters is the
covariance associated with the free (gaussian) groundstate and the other parameter is
induced by the interaction. We would like to stress that the groundstate is annihilated
by Gauss’ law. This holds in particular for the free groundstate. There is therefore no
need of a modified Gauss’ law as suggested in [5].
Thenon-gaussiancharacter ofthegroundstatehassomeunpleasant properties. Trying
to evaluate expectation values directly, using our general expression for the inner product,
onerunsintoveryhardcalculations. However fornumericalcalculationsourinnerproduct
is well suited. To calculate expectation values in a simple way we can use bosonization.
The sequence of deriving various statements of the model is different in this work than
in previous works. We start out by finding the groundstate. This fixes the irreducible
representation within the functional representation that we are using. Having done this,
we derive the anomalous chiral current algebra and find the creation operators and the
structure of the Hilbert space. We also give a proof of bosonization by showing that the
actionof thefermionic operatorsandthecorresponding bosonic ones have thesame action
on the explicit groundstate. Finally we calculate the different 2-point correlators of the
theory.
This is a technical paper. We have tried to keep it readable by putting all proofs and
lengthy calculations in appendices.
1
2 Hamiltonian
The Hamiltonian of massless QED on a circle of circumference L is,
L
Hˆ = dx 1Eˆ2(x) ψˆ†(x)γ(i∂ eAˆ(x))ψˆ(x) , (1)
2 − x −
Z0  
where γ = γ0γ1. In an explicitrepresentation take γ0 = σ , γ1 = iσ and hence γ = σ .
1 2 3
−
One also has the first class constraint operator, Gauss’ law:
Gˆ(x) = ∂ Eˆ(x) e1[ψˆ†(x),ψˆ(x)] = ∂ Eˆ(x) eˆj (x) 0. (2)
x − 2 x − 0 ≈
Furthermore, boundary conditions needs to be specified. We choose,
Aˆ(x+L) = Aˆ(x), (3)
Eˆ(x+L) = Eˆ(x), (4)
ψˆ(x+L) = e−2πiαψˆ(x). (5)
Since Eˆ is periodic, (2) implies that the total electric charge Qˆ = e Ldxˆj (x) vanishes
0 0 0
on physical states. Define the transverse fields,
R
1 L
Aˆ = dxAˆ(x), (6)
T
L
Z0
1 L
Eˆ = dxEˆ(x), (7)
T
L
Z0
x
ψˆ (x) = exp ie dx′Aˆ(x′) iexAˆ +i2παx/L ψˆ(x). (8)
T T
−
 Z0 
With this definition, ψˆ is periodic, ψˆ (x+L) = ψˆ (x). Under agauge transformation
T T T
Λ(x) = eiλ(x), the fields transform as
Eˆ′(x) = Eˆ(x), (9)
1
Aˆ′(x) = Aˆ(x) ∂ λ(x), (10)
x
− e
ψˆ′(x) = Λ(x)ψˆ(x). (11)
Under a small gauge transformation, λ(x) periodic and λ(0) = 0, we find that Aˆ and
T
ψˆ are invariant. Under a global transformation λ(x) = λ(0), Aˆ is still invariant while
T T
ψˆ′ = eiλ(0)ψˆ . Finally, for a large transformation (picking a representive for each class),
T T
λ(x) = 2πnx/L, we have
2πn
Aˆ′ = Aˆ , (12)
T T − eL
ψˆ′ (x) = ei2πnx/Lψˆ (x). (13)
T T
The fields satisfy the following nonvanishing commutators and anticommutators:
[Aˆ(x),Eˆ(y)] = iδ(x y),
−
ψˆ (x),ψˆ†(y) = δ δ(x y). (14)
{ α β } αβ −
2
Let us also introduce the operator Jˆ, which commutes with Gˆ,
Jˆ (x,y) = 1[ψˆ† (x),ψˆ (y)]. (15)
αβ 2 T,α T,β
Using (14) we obtain,
i
[Aˆ ,Eˆ ] = , (16)
T T
L
[Jˆ (x,y),Jˆ (x′,y′)] = δ δ(y x′)Jˆ (x,y′) δ δ(x y′)Jˆ (x′,y), (17)
αβ α′β′ βα′ αβ′ αβ′ α′β
− − −
y x y
[Jˆ(x,y),Eˆ (z)] = e − dx′δ(x′ z) Jˆ(x,y), (18)
L
L − −
(cid:18) Zx (cid:19)
[Jˆ(x,y),Eˆ ] = 0. (19)
T
Note that (17) only holds onall states as long asx = y orx′ = y′. The constraint operator
6 6
now reads,
Gˆ(x) = ∂ Eˆ (x) eˆj (x) = ∂ Eˆ (x) e1[ψˆ†(x),ψˆ (x)] = ∂ Eˆ (x) eJˆ (x,x), (20)
x L − 0 x L − 2 T T x L − αα
ˆ ˆ ˆ ˆ
where E (x) = E(x) E . Hence onthe set of physical states, defined such thatG(x) = 0
L T
on that set, we may−express Eˆ (x) in terms of ˆj (x) = Jˆ (x,x). Doing this one may
L 0 αα
write the Hamiltonian on the set of physical states as,
L L L L
Hˆ = Eˆ2 i dxlim(∂ +ieˆb)γ Jˆ (x,y)+ dx dyV¯(x y)ˆj (x)ˆj (y), (21)
2 T − Z0 y→x y αβ αβ Z0 Z0 − 0 0
where ˆb = Aˆ 2πα and
T − eL
e2L 1 2πnx
V¯(x) = cos . (22)
4π2 n2 L
n>0
X
3 Functional representation
To implement the canonical (anti)commutation relations (14) we will use the functional
Schr¨odinger representation. For fermions we will use the reducible representation first
found in [6].
Let the bosonic operators Aˆ(x),Eˆ(x) and the fermionic operators ψˆ(x),ψˆ†(x) act on
wavefunctionals Ψ(A,η∗,η) of a real bosonic field A(x), a complex Grassmann field η(x)
and its complex conjugate η∗(x) according to
Aˆ(x) A(x), (23)
↔
1 δ
Eˆ(x) , (24)
↔ i δA(x)
1 δ
ˆ
ψ (x) (η (x)+ ), (25)
α ↔ √2 α δη∗(x)
α
1 δ
ψˆ†(x) (η∗(x)+ ). (26)
α ↔ √2 α δη (x)
α
3
A general wave functional may be viewed as an overlap [7] with a product of a bosonic
field state and a Grassman field state, Aη∗η = A η∗η ,
| i | i⊗| i
Ψ(A,η∗,η) = Aη∗η Ψ ,
h | i
Aη∗η A′η′∗η′ = δ(A A′)exp dx η∗(x)η′ (x) η′∗(x)η (x) . (27)
h | i − α α − α α
(cid:20)Z (cid:16) (cid:17)(cid:21)
Here, and in the following, left out integration limits means that the integral should be
taken from 0 to L. Using the field states, the partition of unity and the inner product are
given by functional integration [6, 7],
ˆ1 = DA A A D2η′D2η η∗η η∗η η′∗η′ η′∗η′ , (28)
| ih |⊗ | ih | ih |
Z Z
Ψ Ψ = DAD2η′D2η η∗η η′∗η′ Ψ∗(A,η∗,η)Ψ (A,η′∗,η′). (29)
h 1| 2i h | i 1 2
Z
where we have set D2η = Dη∗Dη.
4 Gauge-invariant states
Gauge-invariant states are annihilated by Gauss’ law,
Gˆ(x)Ψ(A,η∗,η) = 0. (30)
These states are invariant under the transformations that are generated by Gˆ, i.e. the
small and the global gauge transformations. Gauge-invariant wavefunctionals have been
found in [8]. A general gauge-invariant functional is parameterized by a family of distri-
butions denoted f and has the following form (spinor indices are summed over):
Ψ (A,η,η∗) = f(0)(A )+
f T
(cid:20)
∞ 1
+ daxdayf(a)(A ;x ,y ,...,x ,y )η∗(x )γ˜η (y ) η∗(x )γ˜η (y )
a! T 1 1 a a T 1 T 1 ··· T a T a
a=1 Z (cid:21)
X
exp dxη∗(x)M η (x) , (31)
× T T
(cid:20)Z (cid:21)
ˆ
where the real variable A and the Grassman field η (x) are defined in analogy with A
T T,α T
ˆ
and ψ (x), i.e.
T
1 L
A = dxA(x) (32)
T
L
Z0
x
η (x) = exp ie dx′A(x′) iexA +i2παx/L η (x). (33)
T,α T α
−
(cid:20) Z0 (cid:21)
A few comments are in order. The constant matrix M in (31) must satisfy tr M = 0
and M2 = 1. These conditions do not uniquely determine M. However, due to the
4
reducibility of the representation different choices of M give functionals all representing
the same physical state. Furthermore, the matrix M determines the matrix γ˜ through
γ˜ = 1(1 M)γ(1+M). A convenient choice of M giving a gaussian free groundstate is
2 −
M = iγ1. In the following that choice is understood. In the gauge-invariant state (31),
the parameterizing distributions multiply ”base states”,
η∗(x )γ˜η (y ) η∗(x )γ˜η (y )exp dxη∗(x)M η (x) . (34)
T 1 T 1 ··· T a T a T T
(cid:20)Z (cid:21)
Thebasestatesareinvariantundertheexchangeofpairs(x ,y ) (x ,y )butchangesign
i i j j
↔
under exchange of x x or y y separately. We will demand that the distributions
i j i j
↔ ↔
f(a) satisfy
f(a)(A ;...,x ,y ,...,x ,y ,...) = f(a)(A ;...,x ,y ,...,x ,y ,...). (35)
T i i j j T j j i i
It is also worth mentioning that gauge-invariant states characterized by translation-
invariant distributions (in the spatial coordinates) have zero total momentum.
Action of physical operators on gauge-invariant states
Physical (gauge-invariant) operators may be constructed from the operator Jˆ (x,y). Let
αβ
the operator A Jˆ(x,y) = A Jˆ (x,y) act on a general gauge-invariant state parameter-
αβ αβ
·
ized by the family f. The state produced will be a gauge-invariant state parameterized
by a new family, say f , i.e.
A
A Jˆ(x,y)Ψ = Ψ (36)
· f fA
The resulting families f may be expressed in terms of the original family f. In [8] the
A
families f were found for A equal to 1,γ0,γ1,γ respectively. For families having the
A
property (35) the result is
(a)
f (A ;x ,y ,...,x ,y ) =
1 T 1 1 a a
a
δ(x x )f(a)(A ;x ,y ,...,y,y ,...,x ,y )
b T 1 1 b a a
−
Xb=1n
δ(y y )f(a)(A ;x ,y ,...,x ,x,...,x ,y ) , (37)
b T 1 1 b a a
− −
(a) o
f (A ;x ,y ,...,x ,y ) =
γ0 T 1 1 a a
if(a+1)(A ;x ,y ,...,x ,y ,y,x)
T 1 1 a a
−
a
+i f(a+1)(A ;x ,y ,...,x ,x,y,y ,...,x ,y )
T 1 1 b b a a
bX=1n
+f(a−1)(A ;x ,y ,...,x/ ,/y ,...,x ,y )δ(x x )δ(y y ) , (38)
T 1 1 b b a a b b
− −
o
5
(a)
f (A ;x ,y ,...,x ,y ) =
γ1 T 1 1 a a
iδ(x y)f(a)(A ;x ,y ,...,x ,y )
T 1 1 a a
− −
a
+i δ(x x )f(a)(A ;x ,y ,...,y,y ,...,x ,y )
b T 1 1 b a a
−
bX=1n
+δ(y y )f(a)(A ;x ,y ,...,x ,x,...,x ,y ) , (39)
b T 1 1 b a a
−
f(a)(A ;x ,y ,...,x ,y ) = o
γ T 1 1 a a
f(a+1)(A ;x ,y ,...,x ,y ,y,x)
T 1 1 a a
a
f(a+1)(A ;x ,y ,...,x ,x,y,y ,...,x ,y )
T 1 1 b b a a
−
b=1
X
a
+ f(a−1)(A ;x ,y ,...,x/ ,/y ,...,x ,y )δ(x x )δ(y y ). (40)
T 1 1 b b a a b b
− −
b=1
X
These expressions are also valid for a = 0 if sums ranging from one to zero are set to zero.
Inner product and gauge-invariant states
Gauge-invariant state functionals are completely specified by a family of distributions.
The inner product (29), when involving gauge-invariant states only, must therefore be a
mapping from a pair of families to the complex numbers. We will see how that comes
about.
First,considering(29),weobservethatDA = const dA DA ,D2η′D2η = D2η′ D2η
T L T T
and η∗η η′∗η′ = η∗η η′ ∗η′ . Then, after these changes of variables, all dependence
h | i h T T| T Ti
of A has disappeared from the integrand and the integral over A gives just another
L L
divergent constant. The inner product (29) thus becomes
∞
Ψ Ψ = N dA D2η′ D2η η∗η η′ ∗η′ Ψ∗(A ,η∗,η )Ψ (A ,η′ ∗,η′ ). (41)
h g| fi T T T h T T| T Ti g T T T f T T T
Z−∞ Z
On inserting the expression (31) for the two general gauge-invariant functionals, it is
possible to do the fermionic integrals. After dropping the factor N and a factor det(2I)
which emerges in the calculation (I (x,y) = δ δ(x y)) we arrive at the final form of
αβ αβ
−
the inner product on the space of gauge-invariant functionals:
∞
Ψ Ψ = dA g(0)(A )∗f(0)(A ) (42)
g f T T T
h | i
Z−∞ (cid:26)
∞ 1
+ daxdayǫ g(a)(A ;x ,y ,...,x ,y )∗f(a)(A ;x ,y ,...,x ,y ) .
a! i1···ia T 1 i1 a ia T 1 1 a a
a=1 Z (cid:27)
X
We will return to this expression below. Another way of deriving (42) is by making an
appropriateansatzandthenusing thehermiticitypropertiesofthevariousgaugeinvariant
operators to constrain the ansatz as was done in [8].
6
5 Groundstates
Having set up the formalism we are now ready to find eigenstates of Hˆ. Using (37) and
(40) one finds the action of Hˆ on a general gauge-invariant state Ψ (A ,η ,η∗). We have
f T T T
HˆΨ (A ,η ,η∗) = Ψ (A ,η ,η∗), (43)
f T T T f′ T T T
where the family f′ is given, in terms of the family f, by (remember that b = A 2πα)
T − eL
1
f′(a)(A ;x ,y ,...,x ,y ) = ∂2 f(a)(A ;x ,y ,...,x ,y )
T 1 1 a a −2L AT T 1 1 a a
i dxlim(∂ +ieb)f(a+1)(A ;x ,y ,...,x ,y ,y,x)
y T 1 1 a a
− y→x
Z
a
+i dxlim(∂ +ieb) f(a+1)(A ;x ,y ,...,x ,x,y,y ,...,x ,y )
y T 1 1 b b a a
y→x
Z b=1
X
a
i f(a−1)(A ;x ,y ,...,x/ ,/y ,...,x ,y )(∂ +ieb)δ(x y )
− T 1 1 b b a a xb b − b
b=1
X
+V(x ,y ,...,x ,y )f(a)(A ;x ,y ,...,x ,y ). (44)
1 1 a a T 1 1 a a
This expression holds for a = 0 if sums ranging from one to zero are set to zero. The
potential V is defined as
a a
V(x ,y ,...,x ,y ) = V(x y ) [V(x x )+V(y y )], (45)
1 1 a a i j i j i j
− − − −
i,j=1 j>i=1
X X
where
e2L 1 2πnx
¯ ¯
V(x) = 1 cos = 2[V(0) V(x)]. (46)
2π2 n2 − L −
n>0 (cid:18) (cid:19)
X
There is a simple physical interpretation of V. It gives rise to an attractive interaction
between states of different charge, (xy), and a repulsive interaction between states of
equal charge, (xx) or (yy).
Demanding that Ψ (A ,η ,η∗) is an eigenstate of Hˆ with eigenvalue E , i.e.
f T T T
f′(a) = Ef(a), a = 0,1,2,..., (47)
leads by (44) to a complicated set of hierarchy equations coupling different levels (we call
a the level).
It turns out that there are eigenstates that factorize in an electromagnetic (EM) part
and a fermionic (F) part (this is not true if we add a mass term to the Hamiltonian), i.e.
Ψ(A η∗η ) = Ψ(A )Ψ(η∗η ), (48)
T T T T T T
Ψ = Ψ Ψ . (49)
EM F
| i | i ⊗| i
Accordingly their families also factorize,
f(a)(A ,x ,y ,...,x ,y ) = Ψ(A )f(a)(x ,y ,...,x ,y ), a = 0,1,2,.... (50)
T 1 1 a a T 1 1 a a
7
For such states the strategy is: First, split the hamiltonian in an electromagnetic and a
fermionic part defined through Hˆ = 1Eˆ2 + Hˆ and find A -independent eigenstates of
2 T F T
Hˆ with eigenvalue E(A ).
F T
This corresponds to solving (47) ignoring the kinetic term for A in the expression for
T
f′(a) given in (44). Second, having found such eigenstates of Hˆ , (47) and (44) collapse
F
into
1
∂2 Ψ(A )+E(A )Ψ(A ) = EΨ(A ), (51)
− 2L AT T T T T
which is then solved.
We will pursue the strategy outlined above. In Appendix A we show that the states
Ψ (η∗η ), parameterized by the A -independent family f are eigenstates of Hˆ with
N T T T N F
(0)
eigenvalues E (A ). The family is given by f = 1 and for a = 0,
N T N 6
f(a)(x ,y ,...,x ,y ) = Ω (x y ) Ω (x y )Φ(a)(x ,y ,...,x ,y ), (52)
N 1 1 a a N 1 − 1 ··· N a − a 1 1 a a
where
1
Ω (x) = sgn(n+N)eipnx, (53)
N
−L
n
X
a a
Φ(a)(x ,y ,...,x ,y ) = exp ϕ(x y ) [ϕ(x x )+ϕ(y y )] ,(54)
1 1 a a i j i j i j
 − − − − 
i,j=1 j>i=1
and X X 
1
ϕ(x) = p2 +M2 p (1 cosp x). (55)
− np n − n − n
nX>0 n (cid:18)q (cid:19)
Here we have introduced the notation p = 2πn for discrete momenta and the standard
n L
(a)
notation M = e/√π. Note that when e = 0, f is just a product of Ω ’s and the
N N
corresponding functional is gaussian. The eigenvalue E (A ) is, after regularization,
N T
π 2π eA L 1 2
E (A ) = + T N α + p2 +M2 p . (56)
N T −6L L 2π − − 2 − n − n
(cid:18) (cid:19) nX>0(cid:18)q (cid:19)
From the form of the eigenenergy we see that the remaining quantum mechanical system
(51) is a harmonic oscillator with well-known solutions. Denote its lowest energy state by
Ψ (A ) and its eigenvalue by E . Thus,
N T N
2π eA L 1 2
T
Ψ (A ) = exp N α (57)
N T
−ML 2π − − 2 − 
(cid:18) (cid:19)
E = π+ M + p2 +M2 p . (58)
N −6L 2 n − n
nX>0(cid:18)q (cid:19)
Hence the states Ψ (A η∗η ) = Ψ (A )Ψ (η∗η ), parameterized by the family
N T T T N T N T T
ˆ
Ψ (A )f , are eigenstates of H, all with the same energy E = E . These states are the
N T N N
groundstates of Hˆ. The corresponding kets will be denoted Ψ .
N
| i
8
The groundstates Ψ have an important property. Under a large gauge transforma-
N
| i
tion λ(x) = 2πx/L, we have that
2π
A′ = A , (59)
T T − eL
η′ (x) = ei2πx/Lη (x), (60)
T T
which implies that Ψ is mapped into Ψ . Hence a state transforming only by a
N N+1
| i | i
phase under large gauge transformations is the θ-vacuum θ defined as,
| i
θ = e−iNθ Ψ . (61)
N
| i | i
N
X
6 Currents, charges and creation operators
In the last section we found the explicit groundstate(s) of the model. The entire Hilbert
space may now be constructed by the action of physical operators on the groundstate(s).
The different groundstates Ψ all define different representation spaces of the algebra
N
| i
of the physical operators. These representation spaces need not be orthogonal. However,
using a set of well-known creation operators one may construct orthogonal Fock spaces
from the different groundstates. Then, an additional operator connects the different Fock
spaces.
In this section we will define some operatorsandinvestigate their properties within the
functional representation. These operators will be needed when calculating expectation
values in the next section.
We start by defining the currents and their fourier transforms. Let
ˆj (x) = 1 Jˆ(x,x), (62)
0
·
ˆj (x) = γ Jˆ(x,x), (63)
5
·
1
ˆj (x) = 1 ˆj (x) ˆj (x) = ˆj (n)eipnx. (64)
± 2 0 ± 5 L ±
n
(cid:16) (cid:17) X
Hermiticity demands that ˆj†(n) = ˆj ( n). When acting on the representation space(s)
± ±
−
defined by the groundstate(s) Ψ , the chiral currents have a well-known anomalous
N
| i
commutator, a Schwinger term,
1
[ˆj (x),ˆj (y)] = δ′(x y). (65)
± ±
±2πi −
In Appendix B we verify this algebra in the functional representation by calculating its
action on the explicit groundstate Ψ (A η∗η ). In momentumspace the algebra reads
N T T T
[ˆj (n),ˆj†(m)] = nδ . (66)
± ± n,m
±
The chiral charge is defined as,
Qˆ = dxˆj (x). (67)
5 5
Z
9