Spherical gauge fields

15 Pages·0.14 MB·English

by  Bugra Borasoy

Spherical gauge fields
1
Bu¯gra Borasoy2 and Dean Lee3
University of Massachusetts
Amherst, MA 01003
9
9 We introduce the spherical field formalism for free gauge fields. We discuss
9
the structure of the spherical Hamiltonian for both general covariant gauge
1
andradialgauge andpointoutseveral newfeatures notpresentinthescalar
n
field case. We then use the evolution equations to compute gauge-field and
a
J field-strength correlators. [PACS numbers: 11.10.Kk, 11.15.Tk, 11.15.-q]

1
2

1
v
0 1 Overview
0
1
1 Spherical field theory is a new non-perturbative method for studying quan-
0
tum field theory. It was introduced in [1] to describe scalar boson interac-
9
9 tions. The method utilizes the spherical partial wave expansion to reduce
/
h a general d-dimensional Euclidean functional integral into a set of coupled
t
- one-dimensional, radial systems. The coupled one-dimensional systems are
p
then converted to differential equations which then can be solved using stan-
e
h dard numerical methods.4 The extension of the spherical field method to
:
v fermionic systems was described in [2]. In that analysis it was shown that
i
X the formalism avoids several difficulties which appear in the lattice treat-
r ment of fermions. These include fermion doubling, missing axial anomalies,
a
and computational difficulties arising from internal fermion loops. This find-
ing suggests that the spherical formalism could provide a useful method for
studying gauge theories, especially those involving fermions. As a small but
important initial step in this direction, we contribute the present work in
which we introduce and discuss the spherical field method for free gauge
1Support provided by the National Science Foundation under Grant 5-22968 and the
Deutsche Forschungsgemeinschaft
2email: [email protected]
3email: [email protected]
4Forfreefieldtheorytheseequationscanbesolvedexactly,aswewilldemonstratehere
for gauge fields.
1
fields.
The basic formalism for spherical boson fields was described in [1]. In this
paper we will build on those results with most of our attention devoted to
new features resulting from the intrinsic spin of the gauge field. We discuss
the operator structure of the spherical Hamiltonian in detail, using two-
dimensional Euclidean gauge fields as an explicit example. Like standard
field theory gauge-fixing is essential in spherical field theory, and we have
chosen to consider general covariant gauge and radial gauge5. In each case
we derive the spherical Hamiltonian and use the corresponding evolution
equations to calculate the two-point correlators for the gauge field and the
gauge-invariant field strength. Free gauge fields in higher dimensions can
be described by a straightforward generalization of the methods presented
here. Theapplicationofsphericalfieldtheorytonon-perturbativeinteracting
gauge systems and related issues are the subject of current research.
2 Covariant gauge
In this section we derive the spherical field Hamiltonian for general covariant
gauge. We will use both polar and cartesian coordinates with the following
conventions:
~t = (tcosθ,tsinθ) = (t1,t2) = (x,y). (1)
In general covariant gauge the Euclidean functional integral is given by
Ai exp ∞dtL (2)
iD
Z (cid:20)Z0 (cid:21)
(cid:0)Q (cid:1)
where
L = dθt 1F12F12 1 (∂ Ai)2 . (3)
−2 − 2α i
Z
(cid:2) (cid:3)
We can write the field strength F12 as
F12 = 1 ∂ +i ∂ (Ax iAy) ∂ i ∂ (Ax +iAy) (4)
2i ∂x ∂y − − ∂x − ∂y
= 1h(cid:16)eiθ ∂ +(cid:17)i ∂ A+1 e iθ(cid:16) ∂ i ∂(cid:17)A 1 i
− −
√2i ∂t t∂θ − ∂t − t∂θ
5See [3] and referen(cid:2)ces t(cid:0)herein for(cid:1)a discussion of(cid:0)radial gaug(cid:1)e. (cid:3)
2
where
Ax iAy = √2A 1. (5)
±
∓
We now decompose A 1 into partial waves6
±
A 1 = 1 A 1einθ. (6)
± √2π ±n
n=0, 1,
X± ···
Returning to our expression for the field strength, we have
F12 = 1 1 einθ F+1 F 1 , (7)
√2π√2i n − n−
n=0, 1,
X± ··· (cid:0) (cid:1)
where
F+1 = ∂A+n−11 n 1A+1 (8)
n ∂t − −t n−1
F 1 = ∂A−n+11 + n+1A 1 . (9)
n− ∂t t −n+1
We can also express the gauge-fixing term in terms of F 1,
n±
∂ Ai = 1 1 einθ F+1 +F 1 . (10)
i √2π√2 n n−
n=0, 1,
X± ··· (cid:0) (cid:1)
With these changes the Lagrangian, L, is
t F+1 F 1 F+1 F 1 t F+1 +F 1 F+1 +F 1 .
4 n − −n n − n− − 4α n −n n n−
− − − −
n=0, 1, n=0, 1,
X± ···(cid:0) (cid:1)(cid:0) (cid:1) X± ···(cid:0) (cid:1)(cid:0) (cid:1)
(11)
In [1] the spherical Hamiltonian for the scalar field was found by direct appli-
cation of the Feynman-Kac formula. This is also possible here, but in view of
the number of mixed terms (a result of the intrinsic spin degrees of freedom)
we find it easier to use the method of canonical quantization. Let us define
the conjugate momenta to the gauge fields,
δL
π+1 = = t (1 1)F+1 +( 1 1)F 1 (12)
n−1 δ∂A+n−11 2 − α −n − − α −−n
∂t (cid:2) (cid:3)
δL
π 1 = = t ( 1 1)F+1 +(1 1)F 1 . (13)
n−+1 δ∂A−n+11 2 − − α −n − α −−n
∂t (cid:2) (cid:3)
6In our notation A±1 carries total spin quantum number n 1.
n
±
3
Following through with the canonical quantization procedure, we find a
Hamiltonian of the form
H = H , (14)
n
n=0, 1,
X± ···
where
H = 1 π 1 π+1 π+1 π 1 (15)
n −4t −n+1 − n 1 n 1 − n−+1
− − − −
α π 1 +π+1 π+1 +π 1 + n 1A+1 π+1 n+1A 1 π 1 .
− 4t (cid:0)−n+1 n 1 (cid:1)(cid:0)n 1 n−+1 (cid:1) −t n 1 n 1 − t −n+1 n−+1
− − − − − −
Weobtain(cid:0)thecorrespondin(cid:1)g(cid:0)Schr¨odinger(cid:1)timeevolutiongeneratorbymaking
the replacements
A 1 z 1, π 1 ∂ . (16)
±n → n± n± → ∂zn±1
We then find
H = 1 ∂ ∂ ∂ ∂ (17)
n −4t ∂z−1 − ∂z+1 ∂z+1 − ∂z−1
−n+1 −n−1 n−1 n+1
(cid:16) (cid:17)(cid:16) (cid:17)
α ∂ + ∂ ∂ + ∂ + n 1z+1 ∂ n+1z 1 ∂ .
− 4t ∂z−−n1+1 ∂z−+n1−1 ∂zn+−11 ∂zn−+11 −t n−1∂zn+−11 − t n−+1∂zn−+11
(cid:16) (cid:17)(cid:16) (cid:17)
For the sake of future numerical calculations, it is convenient to re-express
H in terms of real variables. Let us define
u = 1 z 1 +z+1 +z+1 +z 1 (18)
n 2 n−+1 n 1 n 1 −n+1
− − − −
v = 1 z 1 +z+1 z+1 z 1 (19)
n 2 (cid:0) n−+1 n 1 − n 1 − −n+1(cid:1)
− − − −
x = 1 z 1 z+1 z+1 +z 1 (20)
n 2i(cid:0) n−+1 − n 1 − n 1 −n+1(cid:1)
− − − −
y = 1 z 1 z+1 +z+1 z 1 (21)
n 2i (cid:0) n−+1 − n 1 n 1 − −n+1(cid:1)
− − − −
for n > 0 and (cid:0) (cid:1)
u = 1 z 1 +z+1 (22)
0 √2 1− 1
−
x0 = √12i(cid:0)z1−1 −z+11(cid:1) . (23)
−
The variables u and y correspond(cid:0)with comb(cid:1)inations of radially polarized
n n
gauge fields with total spin n, while v and x correspond with tangentially
n n
±
polarized gauge fields with total spin n. In terms of these new variables,
±
H = H + H , (24)
0 n′
n=1,2,
X···
4
where
H = 1 ∂2 α ∂2 1 u ∂ +x ∂ , (25)
0 −2t∂x20 − 2t∂u20 − t 0∂u0 0∂x0
(cid:16) (cid:17)
and
H = 1 ∂2 + ∂2 α ∂2 + ∂2 (26)
n′ − 2t ∂vn2 ∂x2n − 2t ∂u2n ∂yn2
(cid:16) (cid:17) (cid:16) (cid:17)
n u ∂ +v ∂ +x ∂ +y ∂
− t n∂vn n∂un n∂yn n∂xn
(cid:16) (cid:17)
1 u ∂ +v ∂ +x ∂ +y ∂ .
− t n∂un n∂vn n∂xn n∂yn
(cid:16) (cid:17)
Despite the somewhat complicated form, we know that the spectrum of H
should resemble that of harmonic oscillators. To see this explicitly let us
define the following ladder operators:
U+ = 1 ∂ + ∂ , U = 1 2α n+αn ∂ + 2+n αn ∂ +u +v ,
n −√2 ∂un ∂vn n− √2 4−(n+1) ∂un 4(n+−1) ∂vn n n
(cid:16) (cid:17) (cid:16) (2(cid:17)7)
V+ = 1 2α n+αn ∂ + 2 n+αn ∂ +u v , V = 1 ∂ ∂ ,
n √2 −4(−n 1) ∂un 4−(n 1) ∂vn n − n n− √2 ∂un − ∂vn
− −
(cid:16) (cid:17) (cid:16) ((cid:17)28)
X+ = 1 ∂ + ∂ , X = 1 2α n+αn ∂ + 2+n αn ∂ +y +x ,
n −√2 ∂yn ∂xn n− √2 4−(n+1) ∂yn 4(n+−1) ∂xn n n
(cid:16) (cid:17) (cid:16) (2(cid:17)9)
Y+ = 1 2α n+αn ∂ + 2 n+αn ∂ +y x , Y = 1 ∂ ∂ ,
n √2 −4(−n 1) ∂yn 4−(n 1) ∂xn n − n n− √2 ∂yn − ∂xn
− −
(cid:16) (cid:17) (cid:16) ((cid:17)30)
for n > 1 and
U+ = 1 ∂ , U = α ∂ +√2u , (31)
0 −√2∂u0 0− √2∂u0 0
X+ = 1 ∂ , X = 1 ∂ +√2x . (32)
0 −√2∂x0 0− √2∂x0 0
We have constructed these operators so that
U ,U+ = V ,V+ = X ,X+ = Y ,Y+ = 1 (33)
n− n n− n n− n n− n
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
5
for n > 1 and
U ,U+ = X ,X+ = 1. (34)
0− 0 0− 0
(cid:2) (cid:3) (cid:2) (cid:3)
All other commutators involving these operators vanish. We can now rewrite
H and H (for n > 1) as
0 n′
H = 1U+U + 1X+X + 2, (35)
0 t 0 0− t 0 0− t
H = n+1U+U + n 1V+V + n+1X+X + n 1Y+Y + 2n+2. (36)
n′ t n n− −t n n− t n n− −t n n− t
We now see that H is equivalent to two harmonic oscillators with level-
0
spacing 1,andH isequivalent to fourharmonic oscillators, two with spacing
t n′
n+1 and two with spacing n 1. It is important to note that H , however, is
t −t 1′
quite different. The s-wave configurations are independent of θ, and so H
1′
does not have terms of the form z+1 ∂ and z 1 ∂ . Furthermore the gauge
0 ∂z+1 0− ∂z−1
0 0
fields are massless and so H depends only on the derivatives of z+1 and
1′ 0
z 1. One consequence of this is that the spectrum of H is continuous, and
0− 1′
calculations involving H and relevant boundary conditions must be done
1′
carefully. We will discuss these issues later in our analysis of correlation
functions.
In preparation for our calculation of gauge-field correlators, let us now
~
couple an external source to the gauge field,
J
L = dθt 1F12F12 1 (∂ Ai)2 +A~ ~ . (37)
−2 − 2α i ·J
Z h i
The new spherical field Hamiltonian is then
H( ~) = H z+1 1 +z 1 +1 . (38)
J − n J−n n− J n
− −
n=0, 1,
X± ···(cid:0) (cid:1)
As noted in [1] the vacuum persistence amplitude is given by
tmax
Z( ~) ∝ lim b T exp dtH( ~) a , (39)
J ttmmianx−→0+h | (cid:20)−Ztmin J (cid:21)| i
−→∞
where a and b are any states satisfying certain criteria. These criteria are
| i | i
that a is constant with respect to the s-wave variables z 1, z+1 and has
| i 0− 0
6
non-zero overlap with the ground state of H as t 0+, and b has non-zero
→ | i
overlap with the ground state of H as t .7 Because the spectrum of
→ ∞
H is continuous, in numerical computations it is useful to include a small
1′
regulating mass, µ, for the gauge fields and then take the limit µ 0.
→
3 Radial gauge
We now derive the spherical field Hamiltonian for radial gauge. We take the
gauge-fixing reference point, ~t , to be the origin,
0
(~t ~t ) A~ =~t A~ = 0. (40)
0
− · ·
We expect this gauge-fixing scheme to be convenient in spherical field theory
calculations for several reasons. One is that non-abelian ghost fields in radial
gauge decouple, as they do in axial gauge. In contrast with axial gauge,
however, radial gauge also preserves rotational symmetry. As we will see, the
spherical Hamiltonian and correlation functions in radial gauge are relatively
simple. Since
~t A~ = t eiθA+1 +e iθA 1 , (41)
− −
· √2
we can impose the gauge-fixing c(cid:2)ondition by setting(cid:3)
A 1 = e2iθA+1. (42)
−
−
With this constraint we express the field strength as
F12 = 1 einθ ∂A+n−11 + 1A+1 . (43)
i√π ∂t t n 1
−
n=X0,±1,··· h i
The radial-gauge Lagrangian is then
L = dθt 1F12F12 (44)
−2
Z
= t (cid:2) ∂A+−1n−1 +(cid:3) 1A+1 ∂A+n−11 + 1A+1 .
∂t t n 1 ∂t t n 1
− − −
n=X0,±1,···h ih i
7One caveathere is that a must lie in a function space overwhich the spectrumof H
| i
is bounded below.
7
We again follow the canonical quantization procedure. The conjugate mo-
menta to the gauge fields are
π+1 = δL = 2t ∂A+−1n−1 + 1A+1 , (45)
n−1 δ∂A+n−11 ∂t t −n−1
∂t h i
and the radial-gauge Hamiltonian has the form
H = 1π+1 π+1 1A+1 π+1 . (46)
4t n−1 −n−1 − t n−1 n−1
n=0, 1,
X± ···(cid:2) (cid:3)
In the Schr¨odinger language the Hamiltonian becomes
H = 1 ∂ ∂ 1z+1 ∂ . (47)
4t∂zn+−11∂z−+n1−1 − t n−1∂zn+−11
n=X0,±1,···h i
As before we now define real variables,
x = i z+1 +z+1 (48)
n √2 n 1 n 1
− − −
vn = √12 (cid:0)zn+11 −z+n1 1(cid:1) (49)
− − −
for n > 0 and (cid:0) (cid:1)
x = i z+1. (50)
0 · 1
−
Our Hamiltonian can be re-expressed as
H = H + H , (51)
0 n′
n=1,2,
X···
where
H = 1 ∂2 1x ∂ , (52)
0 −4t∂x20 − t 0∂x0
H′ = 1 ∂2 + ∂2 1 x ∂ +v ∂ . (53)
n −4t ∂x2n ∂vn2 − t n∂xn n∂vn
h i h i
Let us now define ladder operators
X+ = 1 ∂ , X = 1 ∂ 2x , (54)
n 2∂xn n− −2∂xn − n
8
for n 0 and
≥
V+ = 1 ∂ , V = 1 ∂ 2v , (55)
n 2∂vn n− −2∂vn − n
for n > 0. These ladder operators satisfy the relations
X ,X+ = V ,V+ = 1, (56)
n− n n− n
while all other commuta(cid:2)tors vanis(cid:3)h. I(cid:2)n terms(cid:3)of these operators we have
H = 1X+X + 1, (57)
0 t 0 0− t
H = 1X+X + 1V+V + 2. (58)
n′ t n n− t n n− t
We note the radial gauge constraint has removed the continuous spectrum
from the s-wave sector, and the spectrum of H is purely discrete. Further-
more the splitting between energy levels of H is independent of n.8
n′
~
We now couple an external source to the gauge field,
J
L = dθt 1F12F12 +A~ ~ . (59)
−2 ·J
Z h i
The new Hamiltonian is then
H( ~) = H z+1 1 + +1 . (60)
J − n J−n J n 2
− − −
n=0, 1,
X± ··· (cid:0) (cid:1)
The vacuum persistence amplitude in radial gauge is given by
tmax
Z( ~) ∝ lim b T exp dtH( ~) a , (61)
J ttmmianx−→0+h | (cid:20)−Ztmin J (cid:21)| i
−→∞
where a has non-zero overlap with the ground state of H as t 0+, and
| i →
b has non-zero overlap with the ground state of H as t . Since the
| i → ∞
level-spacing of H diverges as t 0+ only the ground state projection of
1′ →
H at t = 0 contributes, and a no longer needs to be constant with respect
1′ | i
to the s-wave variables z 1. This a rather important point since, as we recall
0±
from [1], the constant s-wave boundary condition at t = 0 follows from the
fact that the value of field at the origin is not constrained. This is however
not true in radial gauge as a result of the gauge-fixing constraint.
8The reason is that in radial gauge the gauge fields are tangentially polarized, and so
purelytangentialexcitationsintwodimensionsdonotcontributetothefieldstrengthF12.
9
4 Gauge-field correlators
In this section we calculate two-point gauge-field correlators using the spher-
ical field formalism. This calculation can be done in several ways, including
numerically. For future applications, however, it is useful to have exact
expressions for use in perturbative calculations (e.g., for evaluating countert-
erms). Herewewillobtainresults bydecomposing thefieldsasacombination
of the ladder operators. Let us start with radial gauge. We have
z+1 = 1 iX+ +iX V+ V (62)
n 1 2√2 n n− − n − n−
−
z+n1 1 = 2√12 (cid:0)iXn+ +iXn− +Vn+ +Vn−(cid:1) (63)
− −
(cid:0) (cid:1)
for n > 0 and
z+1 = i X+ +X . (64)
1 2 0 0−
−
(cid:0) (cid:1)
We would like to calculate the correlation function,
frad (t,t) = 0 A+1 (t)A+1 (t) 0 . (65)
n 1,n′ 1 ′ h | n 1 n′ 1 ′ | irad
− − − −
By angular momentum conservation frad vanishes unless n = n. Also
n 1,n′ 1 ′ −
− −
frad (t,t) = frad (t,t), (66)
n 1,n′ 1 ′ n′ 1,n 1 ′
− − − −
and so without loss of generality it suffices to consider frad for n 0.
For typographical convenience let us define n−1,−n−1 ≥
h{F}t{G}t′i = θ(t−t′)hb|U(∞,t)bFUU((t,t,0′))GaU(t′,0)|ai +θ(t′ −t)hb|U(∞,t′b)GUU((t′,0,t))FaU(t,0)|ai,
h | ∞ | i h | ∞ | i
(67)
where
t2
U(t ,t ) = T exp dtH . (68)
2 1
−
(cid:20) Zt1 (cid:21)
For n = 0, we have
6
frad = 1 iX+ +iX V+ V iX+ +iX +V+ +V .
n 1, n 1 8 n n− − n − n− t n n− n n− t′
− − −
(69)
(cid:10)(cid:8) (cid:9) (cid:8) (cid:9) (cid:11)
10