We present the extension of the Lagrangian loop gauge invariant
representation in such a way to include matter fields. The partition function
of lattice compact U(1)-Higgs model is expressed as a sum over closed as much
as open surfaces. We have simulated numerically the loop action equivalent to
the Villain form of the action and mapped out the beta-gamma phase diagram of
this model.Comment: 10 pages, LaTe

Aroca

Ashtekar

Awada

Azcoiti

Banks

Bukenov

Damgaard

Fort

Gambini

Gianvittorio

Guth

H. Fort

J.M. Aroca

Jansen

Jersak

M. Baig

Mandelstam

R. Siri

Rovelli

Yang

Physics Letters B

English

arXiv

We present the extension of the Lagrangian loop gauge invariant representation in such a way to include matter fields. The partition function of lattice compact U(1)-Higgs model is expressed as a sum over closed as much as open surfaces. We have simulated numerically the loop action equivalent to the Villain form of the action and mapped out the beta-gamma phase diagram of this model

Aroca, J M

Baig, M

Fort, H

Siri, R

CERN Document Server

MATTER FIELDS IN THE LAGRANGIAN
LOOP REPRESENTATION: SCALAR QED
J.M. Aroca1, M. Baig2, H. Fort3 and R. Siri4
1Departament de Matematiques,Universitat Politecnica de Catalunya,
08034 Barcelona, Spain.
2Grup de Fsica Teorica, Universitat Autonoma de Barcelona,
08193 Bellaterra (Barcelona) Spain.
3Instituto de Fsica, Facultad de Ciencias, Universidad de la
Republica. Tristan Narvaja 1674, 11200 Montevideo, Uruguay.
4Instituto de Fsica, Facultad de Ingeniera, Universidad de la
Republica. J. Herrera y Reissig 565, 11300 Montevideo, Uruguay.
Abstract
We present the extension of the Lagrangian loop gauge invariant represen-
tation in such a way to include matter elds. The partition function of lattice
compact U(1)-Higgs model is expressed as a sum over closed as much as open
surfaces. These surfaces correspond to world sheets of loop-like pure electric
flux excitations and open electric flux tubes carrying matter elds at their
ends. This representation is connected by a duality transformation with the
topological representation of the partition function (in terms of world sheets
of Nielsen-Olesen strings both closed and open connecting pairs of magnetic
monopoles). We have simulated numerically the loop action equivalent to the
Villain form of the action and mapped out the -γ phase diagram of this
model. By virtue of the gauge invariance of this description the equilibrium
congurations seems to be reached faster than with the ordinary gauge-variant
descriptions.
1
1 Introduction
The use of loops for a gauge invariant description of Yang-Mills theories may be
traced back to the Mandelstam [1] quantization without potentials. In 1974 Yang
[2] noticed the important role of the holonomies for a complete description of gauge
theories.
In the early eighties a loop based [3] Hamiltonian approach to quantum electro-
magnetism was proposed and generalized [4] in 1986 to include the Yang-Mills the-
ory. This Hamiltonian formulation was given in terms of the traces of the holonomies
(the Wilson loops) and their temporal loop derivatives as the fundamental objects.
They replace the information furnished by the vector potential and the electric eld
operator, respectively. These gauge invariant operators verify a closed algebra (non
canonical) and may be realized on a linear space of loop dependent functions.
Afterwards the loop representation was extended in such a way to include matter
elds: the so-called P -representation [5],[6]. This extension includes open paths in
addition to the closed ones or loops 1.
The loop approach has many appealing features from the methodological point
of view: First, it allows to do away with the rst class constraints of the gauge
theories (the Gauss law). Second, the formalism only involves gauge invariant ob-
jects. This makes it specially well suited to study ‘white’ objects as mesons and
barions in QCD because the wave function will only depend on the paths associ-
ated with the physical excitations. Third, all the gauge invariant operators have a
simple geometrical meaning when realized in the loop space. Additionally, there is
a conceptual motivation: The introduction by Ashtekar [7] of a new set of variables
that cast general relativity in the same language as gauge theories allowed to ap-
ply loop techniques as a natural non-perturbative description of Einstein’s theory.
Furthermore, the loop representation appeared as the most appealing application
of the loop techniques to this problem [8], [9]. Thus, it was realized that this loop
formalism goes beyond a simple gauge invariant description and in fact it provides
a natural geometrical framework to treat gauge theories and quantum gravity.
The non-canonical nature of the loop algebra have made elusive a Lagrangian
loop formalism counterpart of the original Hamiltonian one [10], [11]. In a previous
paper [12] we showed a natural procedure in order to cast loops into the Lagrangian
formalism. In reference [12], it was considered the lattice 4D pure QED model and
1Here we will use the term ‘loop’ for the congurations in presence of matter elds albeit in a
relaxed sense which covers both closed as much as open paths.
2
it was written the action for the loops as a sum over their closed world sheets.
In the present paper we shall continue with the program of setting up the La-
grangian correlative of the Hamiltonian loop formalism. Concretely, we extend this
Lagrangian loop approach in such a way to include matter elds.
We consider the lattice compact version of the scalar electrodynamics SQEDc
which describes the interaction of a compact gauge eld  with the scalar eld
 = jjei’. The self-interaction of the scalar eld is given by the potential (jj2 −
20)
2. For simplicity we shall consider the limit  ! 1 which freezes the radial
degree of freedom of the Higgs eld (it is known that the numerical results obtained
already at  = 1 are indistinguishable from the frozen case). Thus the dynamical
variable is compact, i.e. ’ 2 (−; ]. This model is known to possess three phases,
namely conning, Higgs and Coulomb [13]. The Higgs phase splits into a region
where magnetic flux can penetrate in form of vortices (Nielsen-Olesen strings) and
a region where the magnetic flux is completely expelled [14], the relativistic version
of Meissner eect in superconductivity. Relying on this, we call this two subregions:
Higgs I and II in analogy with superconducting materials.
2 The Loop Representation: Hamiltonian and La-
grangian Formulations
The Hamiltonian loop representation of the scalar compact QED in a lattice { with
sites denoted by x, links denoted by l and plaquettes by p { is given in terms of the
fundamental gauge operators of the P-representation, the ^(P yx ) dened by
2
^(P yx ) = ^
y(x)U^(P yx )^(y) = ^
y(x)
Y
l2P
U^(l)^(y) (1)
(U^(l) are the lattice gauge group operators, ^(x) are the matter eld operators)
and their conjugate momenta: the electric eld operator E^(l) and the ^y(x) and
^(x). They obey the commutation relations
h
E^(l); ^(P yx )
i
= Nl(P )^(P
y
x )
2P yx comprises open as much as closed paths or loopsC at xed t, representing "electro-mesons"
and pure gauge excitations respectively. In this last case ^(C)  W^ (C) i.e. it reduces to the Wilson
loop operator.
3
h
^y(x); ^(P zy )
i
= −ixy^(P
z
y )h
^(x); ^(P zy )
i
= −ixz^(P
z
y ) (2)
where Nl(P ) is the number of times the link l appears in the path P .
The ^(P yx ) operators are the creation operators of the loops i.e.
^(P yx )j0 >= ^
y(x)U^(P yx )^(y)j0 >= jP
y
x > (3)
where j 0 > is the zero loop state (strong coupling vacuum of the system).
In order to cast the preceding loop description in the Lagrangian formalism let
us consider the partition function for the Villain form of the lattice action3 which is
given by
Z =
Z
(d)
X
n
Z
(d’)
X
k
exp(−

2
jj r − 2n jj2 −

2
jj r’− 2k −  jj2); (4)
where we use the notations of the calculus of dierential forms on the lattice of [15].
In the above expression:  = 1
e2
,  is a real compact 1-form dened in each link of
the lattice and ’ is a real compact 0-form dened on the sites of the lattice, r is
the co-boundary operator, n are integer 2-forms dened at the lattice plaquettes,
and k integer 1-forms, and jj : jj2=< :; : >.
If we use the Poisson summation formula
P
n f(n) =
P
s
R1
−1 df()e
2is for
each of the integer variables, the partition function (4) transforms into
Z =
X
s
X
t
Z
(d)
Z
(d’)
Z +1
−1
(d )
Z +1
−1
(d) exp(−

2
jj r − 2 jj2
−

2
jj r’− 2−  jj2)ei2<s; >ei2<t;>: (5)
Integrating in the  and  variables
Z /
X
s
X
t
Z
(d)
Z
(d’) exp(−
1
2
jj s jj2 −
1
2
jj t jj2)
ei<s;r>ei<t;r’−>: (6)
Using the partial integration rule <  ;r >=< @ ;  > (@ = r is the boundary
operator which maps k-forms into (k-1)-forms and where  is the duality operation
3The choice of the Villain form instead of the ordinary Wilson form is only done for simplicity,
with the Wilson action it is also possible to repeat everything of what follows.
4
which maps k-forms into (4-k)-forms) and integrating over the compact ’ and  we
get the constrains (@t = 0) and (@s = t) and thus, we nally arrive to
Z /
X
s
exp(−
1
2
jj s jj2 −
1
2
jj @s jj2) (7)
or
Z /
X
s
exp(−
1
2
< s;
r@ +M2
M2
s >); (8)
where M2 = 

is the mass acquired by the gauge eld due to the Higgs mechanism.
If we consider the intersection of one of the surfaces dened by the integer 2-forms s
(open and closed surfaces) with a t = constant plane we get spatial paths or ‘loops’.
It is easy to show that the creation operator of those paths is just the the ^(P yx )
operator. Repeating the steps from Eq.(4) to Eq.(7) we get for < ^(P yx ) >
< ^(P yx ) >/
X
s
exp(−
1
2
jj s jj2 −
1
2
jj @s+ P yx jj
2): (9)
Thus, we have arrived to an expression of the partition function in terms of the
world sheets of electric string-like congurations: the loop (Lagrangian) represen-
tation. In this representation the matter elds are naturally introduced by means
of open surfaces which are the world sheets of open paths representing the ‘meson-
like’ congurations. In other words, as it is shown in Fig.1, cutting the above open
surfaces with planes t = constant we get the corresponding quantum Hamiltonian
description in terms of open paths P yx (the P -representation).
3 Duality and the topological representation
Another equivalent description of the Villain form is the topological representation
in terms of the topological objects. As our model has two compact variables we
have two topological excitations: monopoles and Nielsen-Olesen strings [16]. The
BKT expression for the partition function of compact scalar QED is obtained via
the Banks-Kogut-Myerson transformation [17] (see Appendix) and is given by
Z /
X
n(m)
exp(−22 < n(m);
M2
@r+M2
 n(m) >) (10)
5
where m = @  n are closed integer 1-forms attached to links which represent
monopole loops and n(m) = n − @  q are integer 2-forms attached to plaque-
ttes corresponding to the world sheets of both Dirac and Nielsen-Olesen strings
(with monopole loops as borders). Thus, comparing (8) and (10) we can observe a
complete parallelism: in both representations we have a sum over surfaces, and in-
tersecting with a plane t = constant we get closed as much as open strings with point
charges at their ends. In the rst case this string-like excitations are ‘electric’ whilst
in the second they are ‘magnetic’. Furthermore, there is a duality transformation
connecting the conning and Higgs II sectors of the phase diagram. We want to
remark that there is a slight dierence between both equivalent descriptions. In the
BKT representation monopoles occur at the ends of both the Nielsen-Olesen strings
(physical excitations) and the Dirac strings (non physical gauge-variant objects) so
we have the corresponding two types of world sheets mixed in the 2-form n(m) of
equation (10). On the other hand the gauge invariant loop description is simpler
and completely transparent from the geometrical point of view.
4 Numerical Analysis
We have performed a standard Metropolis Monte Carlo simulation with the loop
action of (7). We have worked on a hypercubical lattice. Basic variables are integers
n(r) attached to plaquettes p = (r;; ), where r denotes the site and ;  = 1−4
two directions. These variables describe the world-sheets of the loop-excitations.
The loop action of (7) is equivalent to the Villain form. There were not numerical
results of the phase diagram (as far as we know) for the Villain action of SQEDc.
Thus, in order to map out the phase diagram, we made preliminary Monte Carlo
runs along lines of xed  or γ on a 64 lattice. Typically, 5,000 iterations were
done at each coupling and measurements of the specic heat were taken. Then the
coupling was changed by 0:005, and another 5,000 iterations were made, etc. The
resulting crude phase diagram (three lines: conning-Coulomb, conning-Higgs and
Higgs-Coulomb) proved very helpful in guiding our larger scale simulations.
The next step was to localize accuratelly the transition lines in a 84 lattice. With
this aim, we have performed runs of 40,000 iterations after 15,000 termalization
sweeps per-point on the plane  − γ. The phase diagram is shown in Fig. 2.
It is worth while to mention that the loop description avoids the problem of sum-
ming over gauge redundant congurations. This is reflected in the fact that when
simulating the loop action the convergence to the equilibrium congurations seems
to be quite faster than when using the ordinary action in terms of the elds. Fur-
6
thermore, we observed the absence of the strong metastability previously observed
in Monte Carlo analysis of the SQEDc lattice theory with the Wilson action, which
makes very dicult the numerical analysis of the phase transition critical exponents.
This fact is one of the main advantages for using the action dened in eq. (7).
5 Conclusions
First, we have shown how to introduce matter elds in the Lagrangian loop rep-
resentation. It turns out that the partition function of gauge-Higgs theory can be
represented as the sum over world sheets of loops (open and closed).
Second, a correspondence between the loop and the BKT descriptions is patent
and suggests a duality transformation connecting the conning and the Higgs II
regions of the phase diagram. Perhaps this connection found in the lattice also
holds in the continuum, supporting the claims on the existence of a new phase in
QED [18].
Third, simulating numerically a loop-action equivalent to the Villain form of
the SQED model (with frozen radial degree of freedom) we mapped out its phase
diagram. It turns out that it is very similar to the corresponding phase diagram
for the Wilson form [19], [20]. Additionally, these results conrm those previously
obtained for the same model but using the Hamiltonian loop approach [21], because
the mapped phase diagram is qualitatively equal in both cases.
The next step will be the implementation of the Pauli exclusion principle in the
Lagrangian loop approach in order to include fermionic elds. This task has been
accomplished in the context of the Hamiltonian loop formalism in reference [5] where
a transparent geometrical description of ‘full’ QED was given.
Appendix
To obtain the monopole representation (10) we start with the Villain form (4) and
x the gauge ’ = 0. Then, we parameterize the n = rq + n(v), where q run over
arbitrary 1-forms and v over all co-closed 3-forms (rv = 0). n(v) is a solution of
rn = v. If we perform a translation k ! k − q we get the expression
Z /
Z +1
−1
(dA)
X
v
(rv = 0)
X
k
exp(−

2
jj rA− 2n jj2 −

2
jj A+ 2k jj2) (11)
7
where A =  − 2q is a non-compact variable. By shifting A by 2k we nd
dependence on the combination n+rk, which turns to be a solution of rn = v so
we can eliminate the k variable
Z /
Z +1
−1
(dA)
X
v
(rv = 0)
exp(−

2
jj rA− 2n jj2 −

2
jj A jj2) (12)
and performing the gaussian integration we obtain
Z /
X
v
(rv = 0)
exp(−22 < n(v);
M2
r@ +M2
n(v) >) (13)
Performing a duality transformation we get (10) where m = v are now integer
closed 1-forms (@m = 0) and n(m) = n(v) + @  q so @  n = m. It is possible to
express (13) in terms of the k instead of the n variables which reflects the presence
of Dirac and Nielsen-Olesen sheets.
References
[1] S. Mandelstam,Ann. Phys. 19, (1962) 1.
[2] C.N. Yang, Phys. Rev. Lett.33 (1974) 445.
[3] R. Gambini and A. Trias, Phys. Rev. D22 (1980) 1380.
[4] R. Gambini and A. Trias, Nucl. Phys. B278 (1986) 436
[5] H. Fort and R. Gambini, Phys.Rev. D44 (1991) 1257.
[6] R. Gianvittorio, R. Gambini and A. Trias, J.Phys A24 (1991) 3159.
[7] A. Ashtekar, Phys. Rev. Lett. 57, (1986) 2244; Phys. Rev. D36, (1987) 1587.
8
[8] C. Rovelli and L. Smolin, Phys. Rev. Lett. 61 (1988) 1155; Nuc. Phys. B331
(1990) 80.
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[10] D. Armand-Ugon, R. Gambini, J. Griego and L. Setaro, Preprint IF, Jul.1993.
[11] J.M. Aroca and H. Fort, Phys. Lett. B325 ( 1994) 166.
[12] J.M. Aroca, M. Baig and H. Fort, Phys. Lett. B336 (1994) 54.
[13] J. Jersak, in: Lattice gauge theory-a challenge in large scale computing, B.
Bunk, K.H. Mutter, K. Schilling (eds) New York, Plenum Press 1986.
[14] P.H. Damgaard and U.M. Heller, Phys.Rev.Lett. 60 (1988) 1246.
[15] A. H. Guth, Phys.Rev.D21 (1980) 2291.
[16] A.K.Bukenov, A.V.Pochinsky, M.I.Polikarpov, L.Polley and U.J.Wiese, Nucl.
Phys. B (Proc.Suppl.) 30 (1993) 723.
[17] T. Banks, R. Myerson and J.B. Kogut, Nucl. Phys. B129 (1977) 493.
[18] M. Awada and R. Zoller, Nucl. Phys. B365 (1993) 699 and references therein.
[19] K. Jansen, J. Jersak, C.B. Lang, T. Neuhaus and G. Vones, Nucl. Phys. B265
(1986) 129.
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[21] J.M. Aroca and H. Fort, Phys. Lett. B338 (1994) 60.
9
Figure captions
 Figure 1: Scalar QED surfaces as string propagation.
 Figure 2: Phase diagram of scalar QED.
PS files of the figures will be sent by e-mail if
requested to jomaroca@mat.upc.es
10


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