Gauge Fields Supercomputers

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Introduction toQuantum Gauge Theories on Supercomputers

Department of Mathematics and Computer ScienceUniversity of Southern Denmark

Martin Rasmus Lundquist Hansen

August 1, 2012

SupervisorClaudio Pica


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Abstract

Gauge theories currently dominates our description of elementary particles and their

interactions. Starting from the theory of electrodynamics (the simplest of the gauge

theories) I will introduce abelian and non-abelian gauge theories. As a precursor I

will discuss classical field theory and the Lagrangian formulation. I will in particular

introduce Noethers theorem and show the important connection between continuous

symmetries of the Lagrangian and conservation laws. When introducing gauge theo-

ries we discover why the existence of gauge fields and gauge invariance is a necessity

and my work will eventually lead to non-abelian gauge theories and the famous Yang-

Mills Lagrangian. Computation of quantities associated with gauge theories are in

most cases difficult to perform analytically. This leads to the introduction of the path

integral formulation as a modern way to quantise quantum gauge theories.

When all the prerequisites are in place I will introduce numerical techniques to simu-

late quantum gauge theories via a discretised four-dimensional spacetime lattice. The

lattice formulation will enable me to perform simulations of a pure SU(2) gauge theory

using the path integral formulation and the heat-bath algorithm. The simulation will

allow me to extract the string tension (of the static quark-antiquark potential) and

analyse properties such as confinement and asymptotic freedom. These results will be

interpreted in connection with the theory of the strong nuclear interaction, quantum

chromodynamics.

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1Classical ElectrodynamicsIn this section I will derive some fundamental aspects of classical electrodynamics essential

for my study of quantum gauge theories. In the first part of the section I will show how the

Maxwell equations can be expressed using the electric and magnetic potentials and how an

electromagnetic field behave under Lorentz transformation. These calculations will be based onthe differential form of the Maxwell equations.

E = 0

(Gausss law)

B = 0 E = B

t(Faradays law)

B = 0J + 00 Et

(Amperes law)

The derivations serve two equally important purposes. When rewriting the Maxwell equationsin terms of the potentials we will see how gauge invariance is an integral part of the equations.

The importance of this gauge invariance will be clarified in a subsequent section. The second

purpose is to introduce the electromagnetic field tensor and the associated four-vector notation

of electrodynamics.

The second part of the section will be concerned with the Lagrangian formulation of electro-

dynamics and classical field theory in general. We will see how the Maxwell equations can

be derived from the electromagnetic Lagrangian using the Euler-Lagrange equation and how

symmetries and conservation laws are connected using Noethers theorem.

1.1 Potentials and Gauge Invariance

The first thing we need, is to determine the relationship between the electromagnetic field and

the potentials. This can be done fairly easy by analysing the Maxwell equations. Due to

the lack of magnetic monopoles the magnetic field remains divergenceless in all cases i.e. the

magnetic field can be expressed as the curl of the potential.

B = A (1.1)

In the static case the electric field is conservative and we can simply write it as the gradient

of the potential. In the dynamic case we need to use Faradays law to obtain the relationship

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The first restriction implies that a = for some scalar and by applying this result to thesecond restriction we obtain

v +

t

= 0. (1.12)

The second restriction now implies that the term inside the parentheses is a scalar k(t) in-dependent of the spatial coordinates, but it may depend on time. This leads to the following

result.

v +

t= k(t) (1.13)

We could, however, simply absorb the scalar k(t) into since this does not influence the

gradient of either. From this analysis we are now able to construct the gauge invariance of

the potentials.

A A + V V

t(1.14)

In the next section we will see why the existence of this gauge invariance is important. For nowwe will simply use it to adjust the divergence A. There exists a lot of different gauges (choicesfor the divergence), but by choosing the Lorentz gauge we obtain several advantages such as

equal priority in both space and time and symmetry between the two equations describing the

potentials.

A = 00 Vt

(1.15)

By inserting the Lorentz gauge into equation (1.4) and (1.6), respectively, we obtain a much moreappealing result. In the expression below 2 is the dAlembertian generalising the Laplacian to

four-dimensional spacetime.

2V = 00

V

t 2V =

0(1.16)

2A = 00

A

t 2A = 0J (1.17)

The above result is the Maxwell equations expressed using the potentials in a simple and concise

form.

1.2 Lorentz Transformation

I will now show how electric and magnetic fields change under Lorentz transformation. The

derivation of the transformation rules are carried out by considering a capacitor in three different

inertial frames, one at rest and two boosted.

SSS0

vv0

x

y

-

66

-

6

-

--

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changes and we lose the minus sign. This is of course reflected in the transformation rules too.

Ez = (Ez + vBy) (1.26)

By =

By +

v

c2 Ez

(1.27)

The two remaining components Ex and Bx can be accounted for without the use of mathematics.

If we once again reposition our capacitor, this time in the yz-plane, only the distance between

the plates are shortened. However, the charge density is independent of this distance, and the

electric field stays the same. This leads to the simple conclusion that Ex = Ex. We cannot

account for the magnetic field using our capacitor, but we might consider an ideal solenoid

positioned along the x-axis. In this setup the magnetic field is given by Bx = 0ni with n

being the number of turns per unit length and i is the current. Since the length of the coil

is shortened during Lorentz transformation the number n increases, but simultaneously time is

dilated affecting the current. These two factors exactly cancel out such that

Bx = Bx. Thisconcludes our derivation of the transformation rules for electic and magnetic fields, summarised

below.

Ex = Ex, Ey = (Ey vBz), Ez = (Ez + vBy) (1.28)Bx = Bx, By =

By +

v

c2Ez

, Bz =

Bz v

c2Ey

(1.29)

The transformation rules can be expressed in a more compact form using a second order an-

tisymmetic tensor. This electromagnetic tensor contains information about the electromagnetic

field and it transforms in a well-defined way under Lorentz transformation.

F =

0 Ex/c Ey/c Ez/cEx/c 0 Bz ByEy/c Bz 0 BxEz/c By Bx 0

(1.30)

Using the Einstein summation convention, Lorentz transformation of the electromagnetic field

tensor can be calculated asF =

F

, (1.31)

where is the Lorentz transformation matrix. The derivation of the transformation rules used

the fact that electric fields arises from static charges while magnetic fields are due to movingcharges. The distinction between electric and magnetic fields are thus simply a question about

point of view.

1.3 Four-vector Notation

Equation (1.31) uses the so called four-vector notation. Four-vector notation is almost de facto

standard in theoretical physics and I will make extensive use of this notation in the rest of

the thesis. Four-vector notation is an abstract index notation with some fairly simple rules. A

four-vector is nothing but a four-dimensional vector with an index number associated to each

element. The first element in the vector is time related and the remaining three components arespace related. I will use greek letters for indices running from 0 to 3 (elements associated with

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spacetime) and latin letters otherwise. For greek letters the Einstein summation convention

applies when an index is repeated both lowered and raised. For latin letters the Einstein

summation convention is implied as long as any index is repeated.

AA =3

=0

AA (1.32)

In the above example A is a first order tensor (an ordinary four-vector). The result of the implied

sum is the generalisation of the scalar product to four-dimensional spacetime. When using this

notation we have to take the metric signature into consideration. I have chosen to use the metricsignature (-,+,+,+) and hence the metric tensor given by

=

1 0 0 00 1 0 0

0 0 1 00 0 0 1

. (1.33)

From the metric tensor we observe that the zeroth component of any given four-vector will differ

by sign. The metric tensor can futhermore be used to raise and lower indices. This will result

in a change of sign on the zeroth component. For each index that needs to be altered wesimply multiply with the metric signature. Remember the implied sum over repeated indices

when considering the example below.

A = A (1.34)

In this example A is a second order tensor (a matrix). The first index denotes the row and the

second index the column. A tensor is called contravariant if all indices are raised and covariant

if all indices are lowered. This leads to one of the most important features of the four-vector

notation. The implied sum over the product of a contravariant and a covariant tensor is invariant

under Lorentz transformation.AA = invariant (1.35)

Using this information we should now take a closer look at the Maxwell equations in four-vector

notation. I have already stated how the transformation rules are defined in equation (1.31) using

four-vector notation. To express the Maxwell equations in four-vector notation I will need the

dual-tensor associated with F. This dual-tensor is formally defined in the following way.

G = 12

F =

0 Bx By BzBx 0 Ez/c Ey/cBy Ez/c 0 Ex/cBz Ey/c Ex/c 0

(1.36)

In the expression is the Levi-Civita symbol. We are also going to need the four-current

defined as J = (c,Jx, Jy, Jz) with J = v being the ordinary current density. Using four-vector

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notation the Maxwell equations have a very compact form.

F = 0J

(1.37)

G = 0 (1.38)

The second equation is merely a consequence of the first equation and the Bianchi identity.

The use of the dual-tensor is therefore not strictly necessary, but having the explicit form can

be useful from time to time.

F =13

(F + F + F) = 0 (1.39)

One can easily verify that the differential form of the Maxwell equations are contained within

the two four-vector equations. Consider equation (1.37) for = 0 and implied sum over in

explicit form.1

cE

xx +

Ey

y +E

zz

= 0c (1.40)

By rewriting this expression slightly we obtain Gausss law.

E = 0c2 = 0

(1.41)

For = {1, 2, 3} we obtain the three components of Amperes law. For example, using = 1 weobtain the x-component of this vector equation.

1

c2

Ex

t+

Bz

y By

z= B

1

c2

E

t

x

= 0Jx (1.42)

In a similar way equation (1.38) contains the two homogenous Maxwell equations. Gausss law

for magnetism is given by = 0 and the three components of Faradays law using = {1, 2, 3}.

1.4 The Electromagnetic Lagrangian

To connect the Lagrangian formulation of electrodynamics with the previous statement of the

Maxwell equations in four-vector notation, I will derive these equations as the equations of

motion following from the electromagnetic Lagrangian. The Lagrangian consists of two terms

describing the field and the interaction, respectively.

L = Lfield + Lint = 140

FF JA (1.43)

Before I begin the derivation I need the connection between the field tensor and the po-

tential. Together the electric and magnetic potential constitue a four-vector given by A =

(V/c,Ax, Ay, Az). Using this four-potential the field tensor can be expressed as

F = A A. (1.44)

In four-vector notation the gauge invariance of equation (1.14) simply becomes A

A + .

Note how the electromagnetic field tensor is invariant under this transformation due to continuity

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of the second order partial derivatives. In the derivation I will use A as the dynamical variable

leading to the following expression for the Euler-Lagrange equation.

L(A)

L

A= 0 (1.45)

In the next subsection I will show how the Euler-Lagrange equation is derived from the principle

of least action. Before we begin the actual derivation I will rewrite the Lagrangian by lowering

all indices associated with the field tensor F. We will also need to express the electromagnetic

field tensor using the four-potential.

L = 140

FF JA (1.46)

= 140

(A A)(A A) JA (1.47)

= 140

(AA AA AA + AA) JA (1.48)

The first and the last term inside the parentheses are identical and likewise are the two middle

terms, leaving us with a simpler expression.

L = 120

(AA AA) JA (1.49)

We can now consider the derivative inside the parentheses in the Euler-Lagrange equation.

L(A)

= 120

(A)(A)

A + (A)

(A)A (A)

(A)A (A)

(A)A

Because the Lagrangian is linear in all terms each of the above derivatives can be expressedby two Kronecker deltas.

L(A)

= 120

A +

A A A

(1.50)

By applying the rule of index contraction abcb = ac we get rid of the Kronecker deltas. In the

same process we use the metric tensors to raise the indices again.

L(A)

= 120

(A + A A A) = 10

F (1.51)

This was the hard part of the derivation. The second term in the Euler-Lagrange equation is

simply given byL

A= J = J, (1.52)

leading to the final equation of motion

F

= 0J

. (1.53)

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The above result is identical to the inhomogenous Maxwell equations in (1.37) and using the

Bianchi identity we can obtain the two homogenous equations.

1.5 Noethers Theorem

We will now continue our study of the Lagrangian formulation by introducing Noethers theorem.

Noethers theorem states that for every continuous symmetry of the Lagrangian there is a

corresponding conserved quantity called a Noether current. We use the word symmetry for

all transformations leaving the Lagrangian invariant. In this subsection I will try to motivatethe mathematics behind the theorem and in the next subsection I will use the results to derive

a conservation law. I would like to start off with something different, but highly related, namely

the derivation of the Euler-Lagrange equation. This derivation contains an important step we

are going to need when deriving Noethers theorem. The Euler-Lagrange equation is the result

obtained using the principle of least action. If a system evolves over time in (configuration) space

it follows the path that minimises the action. In the ordinary one dimensional case of scalarsthis extremum can be found by setting the derivative equal to zero and solve for the variable in

question. In the case of fields we do something similar using variation of the field. Assuming

the Lagrangian does not depend on higher order derivatives the variation of the action can be

written as

0 = S =

L

+L

()() d

4x. (1.54)

We can rewrite this using the product rule

L

()

= L

() +

L

()

(), (1.55)

such that

S =

L

L()

+

L

()

d4x. (1.56)

The last term in the expression can be turned into a surface integral over the boundary of ourconfiguration space. Assuming the deformation vanish on this boundary the surface integral

is identical zero. This is the important step because it will allow us to change the Lagrangian

by a surface term without affecting the action. The term inside the square brackets has to be

zero given the integral need to vanish for arbitrary and this gives us the Euler-Lagrange

equation.L

L()

= 0 (1.57)

As long as our equation of motion satisfies this expression it will satisfy the principle of least

action. We can now move on to the derivation of Noethers theorem. Consider a transformation

of some field + where is an infinitesimal parameter and is a deformation of thefield. This is a symmetry operation if and only if the equation of motion is invariant under thistransformation. To ensure this, the action needs to be invariant under our transformation, and

we can thus allow the Lagrangian to change by a surface term i.e. by adding a four-divergence

of some J.L L + J

(1.58)

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tensor does only contain contributions from the electromagnetic field and not the interactions

i.e. we should neglect the last term in our Lagrangian (1.43) when inserting into (1.62). In

electrodynamics we take = A and using the result obtained in equation (1.51) we get

T = 10

FA + 14

FF

. (1.63)

The above tensor is not symmetric, but we can construct a new tensor by adding a divergenceless

term K to T where K is antisymmetric in its two first indices. If we use 0K

=

FA we are able to construct the standard stress-energy tensor.

K =

1

0(F

A + FA) (1.64)

From the equation of motion we know that F = 0 when there are no interactions. This

leads to the following expression for the new tensor.

T = T + K = T +

1

0FA

(1.65)

By inserting the definition ofT we obtain the standard expression for the stress-energy tensor.

T =1

0

F(A A) + 1

4FF

=

1

0

FF

+1

4FF

(1.66)

This was the derivation of the stress-energy tensor, but we still need to determine the continuity

equation. If the interacting term in the Lagrangian is zero the conservation law simply readsT

= 0. By allowing interactions the conservation law is less obvious and we have to

calculate the derivative explicitly.

T =

1

0

(F)F

+ FF + 1

2FF

(1.67)

We can use the metric tensors to raise the index on all derivatives.

T =

1

0 (F)F

+ FF + 1

2F

F

(1.68)

The first term can be rewritten using the inhomogenous Maxwell equations, and moved to the

left-hand side. The remaining terms on the right-hand side can afterwards be rearranged into

a more convenient form.

T + JF

=1

20F

F + F + F

(1.69)

The trick is now to use the Bianchi identity F+F+F = 0 to perform a substitution

of the last two terms inside the parentheses.

T + JF =1

20 F

F + F

(1.70)

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The right-hand side is now a product of a symmetric and an antisymmetric factor, and thus, the

entire right-hand side is zero.T

+ JF = 0 (1.71)

The final result is the most general expression for the conservation law of energy and momentum

and we see that T = 0 is a special case arising when there are no interactions. The second

term JF is the Lorentz force density commenly known as f= (E + v B).

To convince ourself that this in fact is the continuity equation describing conservation of energy

and momentum, we should try to write the stress-energy tensor (1.66) in explicit matrix form.

Let us start off by considering the T00 component.

T00 =1

0

00F0F

0 +1

400FF

= 1

2

0E

2 +1

0B2

(1.72)

We recognise this expression as the energy density of the electromagnetic field. Let us now

take a look at the T01 component.

T01 =1

0

11F1F

0 +1

401FF

=

1

0c(ByEz BzEy) (1.73)

This result is the x-component of the Poynting vector (divided by c), defined as the cross product

between the electric and magnetic field.

S =1

0(E B) (1.74)

The Poynting vector is a direct measure of the electromagnetic momentum p = 00S. All theremaining components can be obtained in the same way. By doing so we can write the fullexpression for the stress-energy tensor

T =

1

2(0E

2 + 10

B2) Sx/c Sy/c Sz/c

Sx/c xx xy xzSx/c yx yy xzSx/c zx zy zz

, (1.75)

where ij are the Maxwell stress components, defined by the following expression.

ij = 0EiEj +1

0BiBj 1

2

0E

2 +1

0B2

ij. (1.76)

From this analysis of the components in the tensor we conclude that conservation of the stress-

energy tensor indeed is conservation of energy and momentum for the electromagnetic field.

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2Introduction to Gauge TheoryA gauge theory is a special type of field theory in which the Lagrangian is invariant under

certain transformations. We have already seen how energy and momentum is conserved due to

spacetime translation, and this symmetry exists independent of gauge invariance. In general,

conservation of energy, momentum, and angular momentum are all due to spacetime symmetries.A gauge theory is concerned with another kind of symmetries, namely symmetries that exists

as a consequence of gauge invariance. These symmetries are commenly known as internalsymmetries. A physical field, such as the electromagnetic field, yields a measureable quantity

at each point in space. The underlying potential is different because we are unable to measure

an absolute value, only the difference in potential between to points are measureable. The field

is a realisation of the change in potential and not the potential itself. This leads to the concept

of gauge invariance. As long as we are able to maintain the same change in potential we get

the same physical field.

Each gauge theory is associated with a given symmetry group covering all possible gauge

transformations. In physics we are mainly interested in the unitary groups. The simplest unitarygroup U(1) is an example of an abelian group and the larger SU(n) groups are non-abelian. In

this section I will reveal the importance of gauge fields and the connection to symmetry groups.

It should be noted that from now on I will use natural units = c = 1 in the derivations.

2.1 The Klein-Gordon Field

As an introduction to the concept of gauge theories and the consequence of gauge invariance, I

will present a simple abelian gauge theory based on the complex-valued Lagrangian below.

L = ||2

m2

||2

=

()(

) m2

(2.1)

For simplicity I will treat and as independent fields. This is a somewhat artificial example

but it contains all the properties we need in our explanation. Using the Euler-Lagrange equation

we can obtain the equations of motion for the two fields. These equations will be the well-known

Klein-Gordon equations.

( + m2) = 0 (2.2)

( + m2) = 0 (2.3)

The phase transformation ei

is a symmetry of the Lagrangian. The Lagrangian is, inother words, invariant under the symmetry group U(1) because ei is the generator of elements

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in U(1) using the parameter . To find the conserved quantity associated with this symmetry we

need to determine the deformation of the field . This is done by considering the infinitesimal

form of the symmetry.

+ i (2.4) i (2.5)

Using Noethers theorem, and especially the result obtained in equation (1.60), we can write the

following expression for the conserved Noether current (we have to remember that the Noethercurrent is the sum of both fields).

J = i( ) (2.6)

This is an example of a global symmetry i.e. a transformation independent of position. If we let

= (x) be position dependent we turn the transformation into a local symmetry. This is a muchstronger statement and unfortunately harder to satisfy. The position dependent transformation

ei(x) does not affect the mass term in our Lagrangian, but the derivative will yield aseries of unwanted terms destroying the invariance. The reason for this being that we now

have different transformation rules for each point in space. To solve this problem we need a

way to compare the value of the field at different points in space independent of the phase. By

introducing a new field (the gauge field) we are able to compensate for the phase transformation,

and this will allow us to construct the local symmetry. Assuming the gauge field is given by

some A we can introduce the covariant derivative given by

D = igA. (2.7)The covariant derivative is designed in way that will eliminate the extra terms introduced by

the ordinary differential operator. Assuming that the gauge potential has the following trans-

formation law, we can show that the covariant derivative is independent of local phase rotation.

A A + 1g

(x) (2.8)

If we compare this transformation with equation (1.14) the is equivalent to (x)/g with g

being the coupling constant. The coupling constant is a measure of the strength of the force

exerted in an interaction. In the case of electromagnetism the coupling constant is the chargeof an electron. If we apply gauge transformation and phase rotation on the covariant derivative

simultaneously, we obtain the following transformation.

D

+ ig

A 1

g(x)

ei(x) (2.9)

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Using a little algebra we can show that the covariant derivative indeed is phase independent.

D ei(x) + iei(x)(x) + igAei(x) iei(x)(x) (2.10)= ei(x)(

igA) (2.11)

= ei(x)D (2.12)

The same calculation goes for . Without the gauge field and gauge invariance we would not

be able to construct a local symmetry. This would pose several limitations and the construction

of useful Lagrangians would be limited. This is why gauge fields are a crucial part of several

major field theories.

2.2 Quantum ElectrodynamicsI would like to briefly talk about quantum electrodynamics because it is the simplest real gauge

theory and because conservation of electrical charge is a consequence of phase invariance inthis theory. Quantum electrodynamics describes the interaction between light (photons) and

matter (fermions). Fermions are spin-1/2 particles and in quantum electrodynamics they are

described by Dirac spinors . The associated Lagrangian consists of three different terms.

L = LDirac + LMaxwell + Lint = (i m) 14FF eA (2.13)

The first term is associated with the wave function of the fermion and the second term we recog-

nise as the electromagnetic field. The last term describes the interaction between the fermion

and the electromagnetic field. Assuming the fermion is an electron then e is the electrical charge.

Furthermore one should note that =

0

with

being the Dirac matrices. The Lagrangiancan be written in more compact, but less convenient form using the covariant derivative.

L = (iD m) 14FF (2.14)

The Lagrangian contains the fermion Dirac field and the electromagnetic vector field A, both

with simple equations of motion. For the Euler-Lagrange equation yields

L

()

L

= (i

) + eA + m = 0. (2.15)

By taking the complex conjugate we can turn (which corresponds to the antiparticle) into and by use of the covariant derivative we a get very simple expression.

i eA m = (iD m) = 0 (2.16)

Because we already calculated the equation of motion for the electromagnetic Lagrangian we

can use the result of equation (1.51) in combination with

LA

= e, (2.17)

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to state the second equation of motion for the field A.

F = e. (2.18)

From the asymmetry of the field tensor we can deduce the continuity equation J

= 0 forJ = . This is conservation of electric charge for the current J. The symmetry group of

quantum electrodynamics is still U(1) which means that this Lagrangian is also invariant under ei(x) and the corresponding gauge field is the electromagnetic potential. Using Noetherstheorem one can derive the current J as the Noether current of the global U(1) symmetry. The

covariant derivative has another interesting property. If we consider the commutator relation we

see that it actually defines the associated field tensor.

[D, D] = [, ] + ie[, A] + ie[A, ] e2[A, A]= ie(A A) = ieF

This is a very useful property, because it allows us to easily determine an expression for the

field tensor.

2.3 Yang-Mills Theory

The previous examples was based on the simplest non-trivial symmetry group U(1). The theory

of Yang and Mills extends gauge invariance to non-abelian symmetry groups, providing us with

a larger set of possible Lagrangians. Using our previous knowledge I will try to motivate the

idea behind the Yang-Mills Lagrangian. Let us start off by considering the transformation

exp(iata) (2.19)where a is a parameter and ta are the generators of the symmetry group we want to use. For

SU(n) we will have n2 1 such generators. These generators should satisfy the commutator re-lation [ta, tb] = ifabctc where fabc are the structure constants. In the fundamental representation

they should furthermore be orthonormalised such that tr(tatb) = 12

ab. In infinitesimal form thetransformation reads

(1 + iata) = + iata, (2.20)with 1 being the identity element of the group. I will not write this explicitly in the rest of the

thesis but one should keep in mind that the identity element depends on the group in question.If a is position independent we have a global symmetry and this adds nothing new to our

previous analysis. Using a = a(x) we get the more interesting local symmetry and this willforce us to introduce a gauge field and adjust the derivative. The covariant derivative will be

adjusted to the use of group generators and g is coupling constant of the associated group.

D = igAata (2.21)

We see how the gauge field gets an additional index associated with the group generators.

Using index contraction on a we could simply write A = Aat

a and the notation would be

equivalent to the U(1) case. We can now use the covariant derivative to determine the field

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tensor.

[D, D] = [, ] ig[, Abtb] ig[Aata, ] + g2[Aata, Abtb]=

ig(A

bt

b

A

at

a

igAaA

b[t

a, tb])

= ig(Aa Aa + gfabcAbAc)ta

From this we conclude that the field tensor is given by

Fa = Aa Aa + gfabcAbAc. (2.22)

The electromagnetic field tensor is gauge invariant because the symmetry group is abelian. In

the non-abelian case the field tensor will not be gauge invariant, but we can still write a gauge

invariant Lagrangian. The trace of the field tensor will always be invariant so

L = 1

4FaFa (2.23)

will be a useful Lagrangian. This looks a lot like the electromagnetic Lagrangian except from

the additional index. The full Yang-Mills Lagrangian is simply the Lagrangian of quantumelectrodynamics adjusted to the non-abelian case

L = (iD m) 14

FaFa (2.24)

with the covariant derivative given by equation (2.21). In a pure gauge theory the Lagrangian

is reduced to (2.23) because the only field present is the gauge field itself.

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3Feynman Path IntegralsIn this section I will introduce the concept of Feynman Path Integrals. In the classical limit any

object will follow a single unique trajectory, but in quantum mechanics a particle can follow

any possible path between two points. Imagine the double-slit experiment. In this case the

particle will be able to choose between either slit resulting in two different paths. The particle

will have a given probability of following each of the two paths and the probability of reachingthe destination will be the combined sum of the two paths. This leads to the concept of path

integrals as a sum over an infinite number of possible paths. Even though the particle can

follow any path accessible, it is the paths closest to the classical path, that will give the largest

contribution to the transition amplitude. However, we still need to take all other paths into

consideration if we want the correct probability in the end. In the first subsection I use the

following definition of the path integral in one dimension.

Dx(t) = 1

C()k

dxkC()

(3.1)

Here Dx(t) is an instruction to integrate over all possible paths x(t) and C() is a, so far,unknown constant. This is a discrete version of the path integral where time is sliced into

pieces using a lattice. In the expression denotes the duration between two points in the

time-dimension and x1 to xN denote the points in the space-dimension. This definition results

in zigzag paths when doing the actual computations. To begin this section I will derive the

transition amplitude for a single particle in one dimension and later on I will try to generalisethe concept to arbitrary quantum systems and fields, and show how to calculate expectation

values using the path integral.

Figure 1: This is an example of a single discretised zigzag path between the spatial

points xa and xb in a given timespan T.

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3.1 Transition Amplitude in One Dimension

To begin the introduction of the path integral I would like to use the simplest example i.e. a

particle moving in one dimension. Because the path integral formulation at first sight seems tobe radically different from the Schrdinger formulation, I would like to show how the Schrdinger

equation can actually be derived from the path integral. In order to do so we start with a singleparticle moving in one dimension under the influence of a potential V(x). The action can be

expressed as an integral over the Lagrangian and by discretising time we are able to rewrite

this integral as a sum over all lattice points.

S =

T0

1

2mx2 V(x)

dt =

k

m

2

(xk+1 xk)2

V

xk+1 + xk2

(3.2)

The transition amplitude from xa to xb in some time T is the path integral over the complex

exponential of the action. Formally we define this transition amplitude in the following way (in

this subsection I will write all factors of explicitly).

U(xa, xb, T) = xb| eiHT/ |xa =

Dx(t)eiS[x(t)]/ (3.3)

By inserting the discretised action along with our definition of the path integral we obtain the

following expression for the amplitude. For simplicity we will only consider the very last time

slice explicitly.

U(xa, xb, T) =

dx

C()exp

i

m

2

(xb x)2

Vxb + x

2 U(xa, x , T ) (3.4)As 0 the first term in the exponential will result in rapid oscillations unless xb x. Theserapid oscillations will more or less cancel out and the only contribution is when xb x. Usingthis argumentation I introduce a new small-valued constant such that x = xb + and hencedx = d.

U(xa, xb, T) =

d

C()exp

i

m

2

2

V

xb +

2

U(xa, xb + , T ) (3.5)

We are now going to expand the exponential of the potential around = 0 to first order. We

may as well neglect the small /2 term.

U(xa, xb, T) =

d

C()exp

im2

2

1 i

V(xb)

U(xa, xb + , T ) (3.6)

Under the assumption that the amplitude U(xa, xb + , T ) is a slowly varying function ofx = xb + we may also expand this term around x xb = .

U(xa, xb, T) =

d

C()exp

im2

2

1 i

V(xb)

1 +

xb+

2

2

2

x2b

U(xa, xb, T )

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We are now able to perform Gaussian integration on each term in the integral.

U(xa, xb, T) =

1

C()

2

im

1 i

V(xb)

1 +

i

2m

2

x2b

U(xa, xb, T ) (3.7)

By multiplying the two last parentheses while only keeping terms up to order we get the

following result.

U(xa, xb, T) =

1

C()

2

im

1 i

V(xb) +

i

2m

2

x2b+ O(2)

U(xa, xb, T ) (3.8)

The only way the above expression makes sense in the limit 0, is when the factor in thefirst parentheses is unity. We are thus able to identify our constant.

C() =

2im (3.9)

If we only consider terms of order and multiply by i we are finally able obtain the standard

expression for the Schrdinger equation.

i

TU(xa, xb, T) =

2

2m

2

x2b+ V(xb)

U(xa, xb, T) (3.10)

This result show the close relation to the Schrdinger formulation of quantum mechanics. Just

as the squared norm of the quantum mechanical wave yields the probability, so does the squared

norm of the transition amplitude. Even though the path integral formulation seems to be verydifferent it is nothing but another approach to quantum mechanics.

3.2 Generalisation of the Path IntegralI will now generalise the path integral to an arbitrary quantum system described by coordinates

q = {qi} and conjugate momenta p = {pi} with a given Hamiltonian H(q, p). We begin with thesame equation for the transition amplitude (from now on I will not write factors of explicitly).

U(qa, qb, T) = qb| eiHT |qa

Again we discretise time in slices of = T/N with N being the number of slices. By doing so

the exponential function becomes a product of infinitesimal contributions.

eiHT =

eiHN

(3.11)

The trick is now to use the identity operator to insert a complete set of intermediate states

between each factor of eiH , in the form:

1 =

i dq

ik

|qkqk| (3.12)

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By inserting these factors for k = 1 . . . (N 1) while identifying q0 = qa and qN = qb we maywrite something along the lines of

qb| eiH |qN1qN1| eiH |qN2qN2| eiH |qN3 . . . q2| eiH |q1q1| eiH |qa. (3.13)

For simplicity I skipped the integral term inside the parentheses in the identity operator. In the

limit 0 we may Taylor expand the exponential function.

qk+1| eiH |qk qk+1| 1 iH + O(2) |qk (3.14)

We now see that the Hamiltonian is an operator on the state |qk and we should thus considerwhat terms this operator could contain. The simplest term is merely a function of the coordinates.

In one dimension this would give us something like a| f(x) |b = f(b)a|b = f(b)ab if thesystem is orthonormal. In the general case we get something similar, but the normalization is a

bit different.

qk+1| f(q) |qk = f(qk)qk+1|qk = f(qk)i

(qik qik+1) (3.15)

The result is, in other words, zero unless all coordinates are equal. We can with advantage

express the Dirac delta function in the following way.

(a b) = 12

eip(ab) dp (3.16)

Using the above definition of the Dirac delta function we are able to rewrite the expression.

qk+1| f(q) |qk =

i

dpik2

f

qk + qk+12

exp

ii

pik(qik qik+1)

(3.17)

Next we want to consider a function that only depends on the momenta. This is easier because

the result follows more naturally. Again let us take a look at this example in one dimension.

a| f(p) |b =

dp f(p)a|pp|b =

dp f(p)exp(ip(b a)) (3.18)

We can more or less adopt this result directly as long as we adjust for our normalization.

qk+1| f(p) |qk =

i

dpik2

f(pk)exp

ii

pik(qik qik+1)

(3.19)

The previous two expressions for the coordinates and momenta are almost identical. Under the

assumption that the Hamiltonian only consists of terms that depend on either the coordinatesor the momenta we are able to write a combined expression.

qk+1| H(q, p) |qk =

i

dpik2

H

qk + qk+1

2, pk

exp

ii

pik(qik qik+1)

(3.20)

Sadly, the above expression is not valid in the general case because the Hamiltonian act as

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an operator on the left-hand side. This means the order of the ps and qs are important. If

we assume that the Hamiltonian is Weyl ordered (the ps and qs appear symmmetrically in the

Hamiltonian) we can keep the above expression. It is possible to put every Hamiltonian into

Weyl order using commutation rules and this justifies the use of the expression. Using the result

of equation (3.14) we are able to write the expression below.

qk+1| eiH |qk =

i

dpik2

exp

iH

qk + qk+1

2, pk

exp

ii

pik(qik qik+1)

(3.21)

The above expression is for a single step in time. We need to add an integral over qik and a sum

over k in the exponential function if we want to obtain the complete expression for the transition

amplitude.

U(qa, qb, T) =

i,k

dq

i

kdpik

2

exp

ik

i p

i

k(q

i

k qi

k+1) Hqk + qk+1

2 , pk

(3.22)

Considering the limit 0 we can define the expression below. It should be fairly easy to seethe connection directly.

U(qa, qb, T) =

i

Dq(t)

Dp(t)

exp

i

T0

dt

i

piqi H(q, p)

(3.23)

Assuming we have a non-relativistic particle in one dimension we can show that (3.22) reduces

to equation (3.4), if we take into account that this equation only describes a single time slice.

Our particle will of course have the classical Hamiltonian H = p2/2m + V(q) and in this case

the path integral become

U(qa, qb, T) =

k

dqk

dpk2

exp

ik

pk(qk qk+1) p

2k

2m V

qk + qk+1

2

.

If we use the Gaussian integral given by

exp(bx ax2)dx =

a exp

b2

4a (3.24)we can calculate the integral over pk. This integration will introduce the familiar constant C()

from the previous section and, as expected, the entire expression is similar to (3.4).

U(qa, qb, T) =

1

C()

k

dqk

C()

exp

ik

m

2

(qk qk+1)2

V

qk + qk+12

(3.25)

3.3 The Path Integral for Fields

We have now seen how the path integral is formulated in the case of particles, but we shouldalso consider how we can generalise it to fields. Equation (3.23) is valid for any quantum system

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and hence also for fields. In the case of fields the position is simply replace by the field values(x) and the momenta by the momentum density (x).

q

, p =

L

q =

L

(3.26)

Using these transformations the path integral can be written as

U(a, b, T) =

D

D

exp

i

T0

d4x

H(, )

, (3.27)

where the integral in the exponential function has been turned into an integral over spacetimebecause we use the densities instead. For a scalar field the Hamiltonian density can be written

as

H=

1

22 + ()

2 + 2V() , (3.28)where 2V() = m22 for a free theory. By inserting this Hamiltonian in the path integral wecan perform an integration over the momentum density using (3.24) as we did before.

U(a, b, T) =

D exp

i

T0

L d4x

(3.29)

Here L = 12

()2 V(). This result involves a non-trivial constant, which I have not written

explicitly. However, we are mainly going to use the path integral to obtain expectation values

of observables. In this case the constant is unimportant because the expectation value is a

normalized quantity. The expectation value of a given observable F[] can be calulated as

F =DF[]eiS[]DeiS[] , (3.30)

where S[] is the action of the field. Together with the physical interpretation of the path

integral, this is the important result we are going to use in the numerical computations later on.

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S

EC

T

IO

N

4Lattice Gauge TheoryAll prerequisites needed in a simple lattice simulation are now in place and this section will

be concerned with the techniques of lattice gauge theory. My numerical computations will be

performed using a pure gauge theory based on the simple SU(2) group. Because SU(2) is

non-abelian we will have a Yang-Mills theory, where the group generators are the three Paulimatrices multiplied by one half.

t1 = 12

x =12

0 1

1 0

, t2 = 1

2y =

12

0 ii 0

, t3 = 1

2z =

12

1 0

0 1

In a pure gauge theory (where the only field present is the gauge field) the Lagrangian can be

expressed using equation (2.23) and the group generators.

L = 14

FaFa =

1

2tr

[Fat

a]2

= 1

2tr(FF) (4.1)

In the first part of the section I will discuss how to properly discretise spacetime. This will

be followed by an introduction of the heat-bath algorithm used to perform the actual simula-

tions. Before I present the numerical results I will give a fairly non-technical description of the

properties of an SU(2) gauge theory with connection to the real theory of the strong nuclear

interaction, quantum chromodynamics.

4.1 Discretisation and the Continuum Limit

In lattice gauge theory we discretise spacetime in order to perform computations that would oth-

erwise be difficult to perform analytically. Discretisation of spacetime is not without problems

because we break all spacetime symmetries including Lorentz invariance. Even the number ofgauge invariant objects are reduced by discretisation. With all these disadvantages taken in to

consideration the lattice formulation is still a very powerful tool because it allows us to perform

computations of non-perturbative quantities, and because it is possible to recover the correct

continuum results by taking appropriate limits.

The simulation is performed using a four-dimensional hypercubic lattice with N points in each

direction. To eliminate surface effects I will impose periodic boundary conditions on the lat-

tice. Discretisation of the spatial directions are straight forward and intuitively clear, but the

temporal direction requires an extra step. Gauge theories (quantum field theories in general)

are formulated in the relativistic Minkowski spacetime. Simulations, however, are most easilydone in Euclidean space, so in order to transform Minkowski space into Euclidean space we

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perform a Wick rotation into imaginary time t it. This will intuitively rotate the temporaldirection by 90 degrees on the complex plane, such that we get four orthogonal Euclidean axes.

Discretisation of the temporal direction is now simply a matter of extrapolation from three to

four dimensions. However, the Wick rotation will (among other things) also impact the zeroth

component of the gauge field A0 iA0 and because of this, the Lagrangian in (4.1) will changesign L L. Last but not least, the action S will also change because the integration mea-sure changes d4x id4x. This will change the argument of the exponential function in the pathintegral from purely imaginary to real.

Z =

D eS (4.2)

We can now see that the integrand of the path integral is equivalent to the Boltzmann factor

(with a so far hidden constant). In fact, we will soon discover the close connection to statistical

mechanics. In the previous section we already considered a discretised version of the path

integral. If we want to use the path integral formulation for our lattice we need to determine

a discretised expression for the action of the system. This action should be gauge invariant

and the discretised version of our theory should reduce to the usual Yang-Mills theory in the

continuum limit where the lattice spacing goes to zero. In a pure gauge theory the action will

be given by the integral of the Lagrangian in equation (4.1). The field tensor can be regarded

as the generalised curl of the potential. Using Stokess theorem we know that an integral over a

vector field along a closed contour is equivalent to a surface integral over the curl of the vector

field. From this argumentation we introduce the gauge invariant Wilson loop as an integral over

the potential along a closed contour.

W(C) = exp

ig

C

Adx

(4.3)

In this expression g is still the coupling constant. We will see that the exponential is required

in order to obtain the square of the field tensor in the end. In the discretised case the smallest

closed contours (Wilson loops) are squares connecting four lattice points (see sketch below).

These contours are usually called plaquettes and the sum over all plaquettes will constitute the

action of the system.

r r r r r r

r r r r r r

r r r r r r

r r r r r r

The discretisation of spacetime makes the path integral an ordinary integral, and because the

lattice has a finite extent in in all directions, the path integral becomes finite dimensional. So

naturally, by writing the action as the sum over all elementary plaquettes the path integral

is finite dimensional. I will return to the actual use of the path integral when discussing the

heat-bath algorithm used for the simulations. Instead I will now show that we actually recoverthe usual Yang-Mills theory in the continuum limit when using this action. In order to do so

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we need to introduce the concept of links. The degrees of freedom in a pure lattice gauge

theory will be links between lattice points i.e. each point will be associated with four links to

neighboring points. I will denote the link from site x in the -direction as U(x) and each link

will be an element of our gauge group SU(2). If we traverse a link in the opposite direction we

simply use the inverse element.U(x + ) = U(x)

1 (4.4)

In the case of SU(n) the inverse is nothing but the conjugate transpose. We can define each

link as a discretised version of equation (4.3) above.

U(x) = exp(igaA) (4.5)

In this expression g is the coupling constant and a is our lattice spacing. Since A is an

element of the Lie algebra, U(x) is an element of SU(2). We have in other words used the

index contraction A = Aat

a on the potential where ta are the Pauli matrices mentioned earlier.

For simplicity, the potential A is evaluated half way along the link i.e. at x + 12 . Using thedefinition of our links we can introduce the discretised version of the Wilson loop. In this case

the Wilson loop will be the product of the links encountered around the contour.

-

6

?

Starting from the upper left corner (denote this point x = x) this contour will be given by the

product of the four links encountered around the loop (with proper path-ordering).

W(C) = U(x)U(x + )U(x + )U

(x) (4.6)

Note how I use downwards as the positive direction on the sketch above. Secondly note that

the last two elements are marked with a dagger, because we traverse the links in the opposite

direction. Each plaquette will contribute an action S and the total action is the sum over all

plaquettes.

S =

S (4.7)

Using the mathematical definition of the plaquette we can write the plaquette action S as the

trace over the combined links.

S =

1 12

tr

U(x)U(x + )U(x + )U

(x)

(4.8)

The trace of SU(2) elements will automatically be a real number and the factor of one half is

normalisation of the trace. The leading normalisation factor will be determined later on, such

that each plaquette contribute an action between zero and two. Using the definition of the

group elements (4.5) we can write the following equations for a plaquette in the (, )-plane,

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assuming the spatial coordinates for A are located half way between the points.

U(x) = exp(iagA(x +12

a))

U(x + ) = exp(iagA(x + a + 1

2a))

U(x + ) = exp(iagA(x + a + 12a))U(x) = exp(iagA(x + 12a))

If we assume the potential A is smooth, we can taylor expand around x to first order using

the covariant derivative.

U(x) = exp(iag[A +12

aDA])

U(x + ) = exp(iag[A + aDA +12

aDA])

U(x + ) = exp(iag[A + aDA + 12aDA])U

(x) = exp(iag[A +12aDA])

Most of these terms will simply cancel out and by inserting the remaining terms into (4.8) we

obtain a simple expression.

S =

1 12

tr

exp(iga2(DA DA))

=

1 12

tr

exp(iga2F)

(4.9)

In the above expression we have recovered the field tensor. By expanding the exponential arounda = 0 we can obtain an even more useful expression.

S = 1 1

2

tr 1 + iga2F1

2

g2a4FF (4.10)The term of order a2 does not contribute to the trace because the field tensor is traceless. Thetrace of the identity element is simply 2 and by moving the rest of the constants outside the

trace we obtain

S =

g2a4

4

tr (FF) . (4.11)

If we identity = 4/g2 and approximate the action by a spacetime integral over all plaquettes

we obtain the usual action (the factor of one half is due to symmetry under , exchange).

S =

1

2

tr (FF) d

4

x (4.12)

We have now showed how to discretise spacetime in such a way that we are able to recover the

Yang-Mills theory in the continuum limit.

4.2 The Heat-bath Algorithm

Before we discuss the actual algorithm we should make some important observations. As already

mentioned the path integral will become finite dimensional in the lattice formulation. Using the

already discussed action and the previously determined value of we can write the path integral

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as a product over all plaquettes (the integration measure will be specified in a short while).

Z =

exp(S(U)) dU (4.13)

We now see that the path integral is identical to the partition function, and this will make all

the tools from statistical mechanics available to us. In the lattice formulation the path integral

might be finite, but the number of degrees of freedom is still huge, making the actual computation

impractical. We are instead going to adopt a statistical method for the computations. The pathintegral is a weighted sum over all possible path configurations, but the Boltzmann factor will

suppress a very large percentage of these. In fact, we only need a small number of configurations

to determine the dynamics of the system, as long as these configurations are weighted according

to the Boltzmann factor. To simulate these configurations we are going to use the heat-bath

algorithm. This algorithm was originally designed for the Ising spin model, but due to the path

integrals connection to statistical mechanics we can adjust it to a gauge theory as well. Usingthis algorithm we successively update each link using a heat- bath at inverse temperature in

order to thermalise the system. Once the system is in equilibrium we can obtain expectation

values as shown in (3.30) by simply averaging over all M configurations, because the exponential

weightning will be build into the algorithm.

F =DU F[U]eS[U]DUeS[U] 1M

Mn=1

F(Un) (4.14)

In this way we are able to calculate the expectation value of any gauge invariant quantity.

In the simulations I will mainly be interested in the expectation value of Wilson loops. ForWilson loops we have M = 6N4 configurations because each lattice point is associated with

six plaquettes. From the above expression we are able to conclude that the statistical error will

decrease as 1/

M. We are in other words going to need a fairly large lattice to reduce the

error.

Let us now continue with the construction of the heat-bath algorithm. Before the algorithm

starts we need some initial configuration. I have chosen to use an ordered starting configuration

with all links set to identity (cold start). To bring the system into equilibrium we replace each

link (one at a time) with a new link chosen randomly from all elements in the SU(2) group, with

exponential weightning according to the Boltzmann factor in the path integral.

dP(U) exp(S(U))dU (4.15)

Here P is the probability density for choosing some link U. Replacement of all links in the

lattice will constitute a single iteration of the algorithm and the next step in a Markov chain of

configurations. All elements of SU(2) can be parameterised using the three Pauli matrices and

the identity matrix.

U = a0I + ia1x + ia2y + ia3z = a0 + ia3 a2 + ia1

a2 + ia1 a0

ia3

(4.16)

Here we assume that a is real four-vector of unit length. The SU(2) group is isomorphic to the

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four-dimensional unit sphere S3 (there is a one-to-one correspondance between points on the

sphere and elements in the group). Finding elements of SU(2) will in other words be equivalent

to calculation of points on the four-dimensional unit sphere. Using this argumentation we

introduce the invariant Haar measure given by

dU =1

22(a2 1)d4a. (4.17)

When updating a given link we only need to consider the contribution to the action given by the

six plaquettes containing the given link. In fact, for each of the six plaquettes in question we

need the product of the three remaining links in order to determine a new value for the link we

want to update. The product of such three links are called a staple due to their shape. There

will be six staples, two in each of the three directions orthogonal to the link.

6

-?

-

?

The above sketch show the two staples in the (, )-direction associated with the black link

in the middle. The backwards staple is the blue one and the forwards staple is the red. If wedenote the staples by Us for s = 1 . . . 6 we can write (4.15) as

dP(U) exp12TrU6

s=0Us

dU, (4.18)

where U is the link we want to update. Because SU(2) elements have positive determinant we

can fairly easy conclude that any sum of SU(2) elements is proportional to another element of

SU(2) with the square of the determinant being the factor of proportionality.

6s=0

Us = kU , k = det

6

s=0

Us

1/2(4.19)

Using this property we can rewrite (4.18) into a more suitable expression.

dP(UU1) exp 12

kTr(U)

dU =1

22(a2 1)exp(ka0)d4a (4.20)

To generate a new element for our link we simply have to determine points on the four-

dimensional sphere with exponential weightning towards a0 and the three remaining points

chosen randomly (but with appropriate norm). Following this prescription we simply replace the

original element with U = UU1 where U is the element we need to generate. We can rewrite

the Dirac delta function in (4.20) into a more explicit expression and this will help us determine

an algorithm for selecting a0 appropriately.

(a2 1) exp(ka0) = 12da0d(1 a20)1/2 exp(ka0) (4.21)

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Here d = sin dd is the differential solid angle of the remaining three components ofa and

together they should constitute a vector of length (1 a20)1/2. Now we simply have to determinea value for a0 stochastically in the interval 1 < a0 < 1. We already know the probabilitydensity given by

P(a0) = (1 a20)exp(ka0), (4.22)so we simply choose a random number x uniformly distributed in the interval exp(2k) < x < 1.Using the random number we can now determine a trial a0 in the following way.

a0 = 1 +ln(x)

k(4.23)

We accept this trial number with probability (1 a20)1/2 to correct for this factor. If the trial a0 isrejected we simply pick another random number and try again. The remaining three components

can be determined using two random numbers x and y in the interval (0, 1) in combination with

the parameterisation of the sphere in three dimensions with radius r = (1 a20)1/2.a1 = r sin(x) cos(2y)

a2 = r sin(x) sin(2y)

a3 = r cos(x)

This concludes the heat-bath algorithm. For my simulations I have written an implementation

in C++ that enables me to calculate Wilson loops of different sizes, depending on the value of

. The source code can be found in Appendix A.2.

4.3 Properties of an SU(2) theory

Before we review the results of the simulation we should consider some of the properties as-

sociated with our gauge theory. An SU(2) theory will have two color charges. The real world

theory of the strong nuclear force, quantum chromodynamics, is based on the larger SU(3) group

where we have three color charges (usually denoted red, green, and blue). In QCD a quark can

take on any of the three colors and quarks can be combined into mesons and baryons as long asthe color combination is neutral. A green quark can for example be combined with an antigreen

quark (an antiparticle) in order to form a meson. The SU(2) theory will act in similar way, but

with only two colors.

There are two interesting properties that apply to non-abelian gauge theories in general i.e.

confinement and asymptotic freedom. These properties can be explained using the static quark-

antiquark potential. The static potential is the limit where the quarks are assumed to be infinitely

heavy, point-like color sources. This is more or less equivalent to the use of a test charge in

electrodynamics. The static quark-antiquark potential can be approximated by

V(r) = r

+ r, (4.24)

where is some constant and is called the string tension. The figure below show this potential

as a function of distance.

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Figure 2: The static quark-antiquark potential as a function of distance.

For small distances we observe asymptotic freedom i.e. the static quark-antiquark potential

becomes asymptotically weaker as the distance decrease. Because of asymptotic freedom quarks

in very close proximity will appear as if they were free particles. For large distances we observethe opposite effect, because the static quark-antiquark potential rises linearly with distance.

This phenomenon is known as confinement and it is the reason quarks cannot exist as freeparticles. Instead they will combine into mesons and baryons. If one tries to pull two quarks

apart the gluon flux tube connecting the two quarks will at some point break and the energy

stored in the flux tube will manifest in a new quark pair and we will now have two mesons as

shown in the sketch below.

} } - } } + } }

This is where the physical interpretation of the Wilson loop becomes extremely important. A

particle traversing spacetime will gain a path dependent phase factor due to its interaction

with the gauge field. This phase factor is the Wilson loop defined in (4.3). Now, the action ofthis phase factor can be seen as an elementary excitation of the field located on the loop. This

elementary excitation can be interpreted as the creation of a static quark-antiquark pair which

will annihilate again after a period of time.

q q

Using this interpretation of the Wilson loop we will actually be able to extract the string tension

associated with a static quark-antiquark pair. Because the potential energy depends linearly

on the distance for large seperations, the Wilson loop is expected to decrease as a function of

the area.W(C) = exp(A + O(p(C))) (4.25)

Here is the string tension and A is the area of the Wilson loop. The error will be propertional

to the perimeter of the loop p(C). If we compared with the analytical expression for the Wilsonloop (4.3) we clearly expect the string tension to depend on . This is not an exact expression but

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it works fairly well for small to moderate values of beta. The area can be expressed as A = a2S2

where a is the lattice spacing and S is the side length (assuming the loops are squares). We

will be able to measure the dimensionless quantity a2 as a function of the Wilson loop.

a2 = ln W(C)S2 + O(p(C))S2 (4.26)

From now on I will refer to a2 as the string tension. Without going into detail about renor-

malization and the continuum limit we are still going to need a connection between the lattice

spacing and the coupling constant. This connection can be written as a2 exp(k) for someunimportant constant k. We are merely interested in the exponential drop-off as a function of because we are going to need it when interpreting some of the results. From our knowledge

of this exponential drop-off, equation (4.25) will automatically be valid in the strong limit where 0, because the area is dominant. However, in the weak limit the equation will onlybe valid for (infinitely) large Wilson loops. I will address this problem in a short while. Using

the numerical techniques previously described we are now going to see how the theory behaves,

and whether or not we can confirm the existence of confinement and asymptotic freedom.

4.4 Numerical Results

The simulations are performed using 244 lattice points and carried out on an ordinary laptop. The

lattice size is chosen as a balance between the required computational time and the statistical

error associated with the observables. To obtain my results I used 200 iterations of the heat-

bath algorithm to thermalise the system and afterwards I performed another 200 iterations while

simultaneously calculating observables. In the end I averaged over all measurements to obtain

the final result. The only observable I have calculated is various sizes of the Wilson loop.The simplest Wilson loop, the plaquette, is a direct measure of the action in the system. The

expectation value of the plaquette P is given by the normalised sum over all plaquettes.

P = 16N4

12

tr(U) (4.27)

We should notice that the average of the action density is given by S = 2(1 P). If wetake a look at the calculated values in figure 3 (on the next page), we clearly see how the

points follow the analytical values very precisely, in both the weak and strong coupling limit.

We furthermore have a smooth transition between the two limits.The analytical expression for the strong and weak coupling limits can be calculated because the

average plaquette value will be equivalent to the internal energy in the corresponding statis-

tical mechanical system. From statistical mechanics we know how to calculate the expectationvalue of the internal energy using the partition function. We can adopt this result directly when

replacing the partition function with the path integral.

P = 16N4

ln Z

(4.28)

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Figure 3: The average plaquette value as a function of. The analytical values inthe weak and strong coupling limit are plotted as well.

The actual calculation of the expectation value in the two limits is not trivial, but the values are

known to be as listed below.6

P = 4

(Strong coupling limit)

P = 1 34

(Weak coupling limit)

Because the standard deviation for the plaquette is of order 104 the error bars would not be

visible on the plot, and I have therefore not added them. The standard deviations can be found

in the appendix.

We should also consider what happens for larger Wilson loops. If the Wilson loop is comparable

to the lattice size we would expect deviations caused by the finiteness of the lattice. I have only

considered (square) Wilson loops up to 6 6 so the finiteness should not be a problem. These

results are shown in figure 4 on the next page. For the same reason mentioned before, I havenot added error bars on this plot either.

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Figure 4: The average value of different sized Wilson loops as a function of. Someknown analytical values in the weak and strong coupling limit are plotted as well.

We see that the average value of the Wilson loop decrease as a function of the loop side.

Considering the connection between the Wilson loop and its area (4.25) this is expected. In the

strong coupling limit we expect the following analytical expression

P = 4

S2, (4.29)

where S is the side length of the Wilson loop. This behaviour is visible for the 1 1 and 2 2Wilson loops, but for larger loop sides this value is approximately zero. In the weak coupling

limit we expect the values to asymptotically approach unity as an inverse function of . This

behaviour does also seem to be correct. For less than 2 the expectation value of large Wilsonloops is significantly smaller than the standard deviation (of order 104). This causes some of

the obtained values to be slightly negative, but well within range of the statistical error.

The string tension is obtained in the limit where the Wilson loop becomes infinitely large.

Luckily, in practice we can use some relatively small Wilson loops. In the strong couplingregime I am unable to obtain consistent values for Wilson loops larger than 2 2. Fortunately,the loops of size 1 1 and 2 2 both yields the expected value using equation (4.25). When wereach the transition between the strong and the weak coupling regime the equation no longer

yields the expected result. In this case we expand the equation with lower order terms in the

exponent.W(C) = exp(A BS CS2) (4.30)

In this equation we identify C = a2. When we reach the transition around = 2 the values

for Wilson loops up to 6 6 begin to become fairly consistent. By fitting these loops to theequation above we can extract a more precise value for the string tension.

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Figure 5: Examples of curve fitting to Wilson loops for various values of .

Figure 5 show the fitting of Wilson loops to equation (4.30) for various values of in the

transition area. We see that the curves intersect the points very precisely in all cases. This

should enable us to obtain a useful value for the string tension.

Figure 6: The extracted values for the quantity a2 as a function of.

We now have the values for the string tension in figure 6 above. The horizontal line is the strong

coupling limit and the vertical line is the weak coupling limit. These are given by the analytical

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expressions below.

a2 = ln

4

(Strong coupling limit)

a2 = exp62

11( 2) (Weak coupling limit)

The expression for the weak coupling limit is not a true analytical equation, but an estimated

fit derived by Creutz.6 Up to and including = 1.6 the points are plotted using the pure area

relation (4.25) for the Wilson loop. The remaining points are plotted using the fitting method. In

the strong coupling limit the points lie very close to the expected analytical values, and when we

reach the transition to the weak coupling limit the points being to decrease correctly. However,

when we reach 2.5 the points starts to diverge from the expected analytical value again.As already mentioned, the lattice spacing has an exponential drop-off as a function of . In the

weak coupling limit this causes the perimeter of the Wilson loop to become more dominant thanthe area, and this is why equation (4.25) no longer remains valid. To circumvent this problem

Creutz proposed to use the ratio between Wilson loops to calculate the string tension.5

(I, J) = ln

W(I, J)W(I 1, J 1)W(I, J 1)W(I 1, J)

(4.31)

In the above expression W(I, J) is a rectangular Wilson loop of dimension I J. The ratiois constructed in such a way that the Wilson loops in the nominator and denominator has the

same total perimeter and cancel out. This should allow us to obtain a much more precise value

for the string tension in the weak coupling limit. A simple calculation using (4.25) show thatthe expression inside the logarithm is equal to exp(a2) and this yields = a2. Because Ialready calculated the square Wilson loops I will use I = J in my calculations.

Figure 7 (on the next page) show the extracted string tension using the Creutz ratio. In thestrong coupling limit we obtain the same result as before, but in the weak coupling limit the

obtained values are much more precise. The plot is created by gradually using larger Wilson

loops (as compensation for the decreasing lattice spacing) to extract the string tension. In the

strong limit I use I = 2 and in the weak coupling limit I use I = 3 . . . 5.

I have tried to estimate some error bars for the calculated values on this plot. When performing

the simulation I only stored the observables (i.e. the value of the Wilson loops) and the cor-responding standard deviation. To estimate the statistical error I used a sort of bootstrapping

method to resample the original data series. I constructed a Gaussian distributed data set for

each Wilson loop using the known mean value and standard deviation. From these data sets

I calculated the string tension and used the standard deviation (of the string tension) as an

estimate of the error. This method is not perfect because the correlation between observables is

lost. It should, however, give a reasonable estimate of the statistical error. One should further-more note that the logarithmic scale on the figure distorts the error bars (the lower and upper

error bars do have the same magnitude).

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Figure 7: The extracted values for the quantity a2 as a function of using theCreutz ratio.

To give a physical interpretation of the plot we need to remember that small values of cor-

responds to large distances and large values of corresponds to small distances. Using this

interpretation of we can argue that confinement seems to be present in the strong coupling

limit and asymptotic freedom in the weak coupling limit. This show that an SU(2) gauge theory

indeed admits confinement as well as asymptotic freedom, just as quantum chromodynamics.

Figure 8: All the extracted values for the quantity a2 using different loop sizes.

Figure 8 show all the values of the string tension for the different choices of I. We see that allWilson loops diverge from the weak coupling limit once the value of is large enough.

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Conclusion

From the theory of classical electrodynamics we derived and reviewed several important results.

We derived the gauge invariance of the electromagnetic potential, and later on we observed that

the existence of gauge invariance was a cornerstone in the construction of a local U(1) symmetry.

As an introduction to the Lagrangian formulation we used the electromagnetic Lagrangian and

derived the Maxwell equations as the equations of motion. By introducing Noethers theorem

we were able to determine the relationship between continuous symmetries of the Lagrangian

and corresponding conservation laws. As an example we used Noethers theorem to show the

connection between the symmetry of spacetime translation, and the conservation of energy and

momentum for the electromagnetic field.

Following the introduction of various concepts in classical field theory we continued with aintroductory study of gauge theories. We showed that gauge theories are concerned with in-

ternal symmetries, as opposed to spacetime symmetries considered in the previous section. We

considered the internal symmetry of phase rotation of a complex scalar field. We saw that thesymmetry of global phase rotation was an intrinsic property of the Lagrangian. However, if we

wanted to transform the phase rotation into a local symmetry we had to introduce the gauge

field and exploit the existence of gauge invariance. Just as spacetime symmetries resulted in a

conserved quantity, so do (global) internal symmetries. We argued that conservation of elec-

trical charge was a result of a global U(1) symmetry in quantum electrodynamics. From these

preliminary studies of abelian gauge theories we expanded the concept to non-abelian groups.

This lead to the famous Yang-Mills theory and the corresponding Lagrangian.

To perform our numerical computations we needed a way to discretise gauge theories. This

lead to the introduction of the path integral formulation. The path integral formulation providesa way of calculating quantum amplitudes and expectation values. Because the path integral

formulation at first sight seems to be radically different from the Schroedinger formulation, we

started off by showing that the Schroedinger equation can actually be derived from the definition

of the path integral. After this introduction we derived a general expression for the path integral

and showed how to rewrite this expression for scalar fields.

Using the previously obtained results we were now able to introduce the concept of lattice

gauge theory. To perform our numerical computations we discretised spacetime using a four-

dimensional hypercubic lattice. We saw how this discretisation was performed using Wilson

loops, in way that allowed us to obtain the correct Yang-Mills theory in the continuum limit.

The discretisation of spacetime turned our path integral into a finite dimensional ordinary in-tegral, but the number of degrees of freedom was still huge. Fortunately, the path integralwas now identical to the partition function used in statistical mechanics. From the connection

to statistical mechanics we were able to adopt the heat-bath algorithm originally designed for

the Ising spin model. When the algorithm was in place we were finally able to perform our

numerical computations. We performed computations of various sized Wilson loops. From the

interpretation of the Wilson loop as the creation of a static quark-antiquark pair, we were able

to extract the associated string tension. Using this string tension we observed that a pure SU(2)

gauge theory indeed admits confinement as well as asympototic freedom.

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Bibliography

[1] David J. Griffiths, Introduction to Electrodynamics, Prentice Hall, 3rd Edition, 1999.

[2] John D. Jackson, Classical Electrodynamics, Wiley, 2nd Edition, 1975.

[3] Michael E. Peskin and Daniel V. Schroeder, Introduction to Quantum Field Theory, Westview

Press, 1st Edition, 1995.

[4] Claude Itzykson and Jean-Bernard Zuber, Quantum Field Theory, McGraw-Hill, 1st Edition,

1980.

[5] Michael Creutz, Quarks, Gluons and Lattices, Cambridge University Press, 1st Edition, 1983.

[6] Michael Creutz, Monte Carlo study of quantized SU(2) gauge theory, Physical Review D 21,

2308, 1980.

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Appendix

This appendix contains the obtained numerical results for the Wilson loops and the source code

for the program used to perform the numerical calculations.

A.1 Numerical Values for the Wilson Loops

Data set for rectangular Wilson loops.

B e t a W i l s o n 2 x 1 W i l s o n 1 x 2 W i l s o n 3 x 2 W i l s o n 2 x 3 W i l s o n 4 x 3 W i l s o n 3 x 4 W i l s o n 5 x 4 W i l s o n 4 x 5 W i l s o n 6 x 5 W i l s o n 5 x 6

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 . 2 0 . 0 0 2 4 8 1 1 2 0 . 0 0 2 4 8 7 8 9 0 . 0 0 0 0 2 0 1 3 - 0 . 0 0 0 0 2 4 8 0 - 0 . 0 0 0 0 1 0 5 5 - 0 . 0 0 0 0 0 6 0 2 0 . 0 0 0 0 1 7 7 3 - 0 . 0 0 0 0 5 8 4 5 - 0 . 0 0 0 0 1 6 5 9 0 . 0 0 0 0 3 2 6 1

0 . 4 0 . 0 0 9 8 3 7 7 1 0 . 0 0 9 8 9 3 2 3 0 . 0 0 0 0 0 4 0 3 0 . 0 0 0 0 0 6 3 0 0 . 0 0 0 0 3 5 2 8 0 . 0 0 0 0 0 6 0 6 - 0 . 0 0 0 0 1 3 7 3 - 0 . 0 0 0 0 4 5 5 8 0 . 0 0 0 0 0 3 2 8 0 . 0 0 0 0 1 4 8 2

0 . 6 0 . 0 2 1 9 6 0 2 7 0 . 0 2 1 9 3 3 2 2 - 0 . 0 0 0 0 4 1 9 1 - 0 . 0 0 0 0 0 5 6 1 0 . 0 0 0 0 2 2 4 6 0 . 0 0 0 0 0 6 9 7 - 0 . 0 0 0 0 2 2 7 0 - 0 . 0 0 0 0 1 4 7 8 - 0 . 0 0 0 0 1 0 3 0 0 . 0 0 0 0 2 5 6 2

0 . 8 0 . 0 3 8 4 0 4 1 6 0 . 0 3 8 4 2 1 9 9 0 . 0 0 0 0 5 4 7 1 0 . 0 0 0 0 6 1 3 6 0 . 0 0 0 0 4 4 8 1 0 . 0 0 0 0 1 2 8 3 0 . 0 0 0 0 1 2 5 1 - 0 . 0 0 0 0 0 7 6 7 - 0 . 0 0 0 0 1 5 5 0 - 0 . 0 0 0 0 1 0 8 9

1 . 0 0 . 0 5 9 1 8 8 9 3 0 . 0 5 9 1 6 7 4 6 0 . 0 0 0 2 4 3 2 3 0 . 0 0 0 2 0 1 3 5 - 0 . 0 0 0 0 3 4 0 2 0 . 0 0 0 0 0 2 6 9 - 0 . 0 0 0 0 2 4 3 2 - 0 . 0 0 0 0 3 6 2 1 0 . 0 0 0 0 5 3 5 3 - 0 . 0 0 0 0 5 5 6 8

1 . 2 0 . 0 8 4 2 7 1 9 3 0 . 0 8 4 2 4 7 0 0 0 . 0 0 0 5 8 8 4 1 0 . 0 0 0 6 1 1 8 7 0 . 0 0 0 0 0 5 0 5 - 0 . 0 0 0 0 2 8 8 0 0 . 0 0 0 0 2 4 1 1 0 . 0 0 0 0 0 5 8 9 0 . 0 0 0 0 1 9 1 3 0 . 0 0 0 0 0 5 8 2

1 . 4 0 . 1 1 4 5 3 5 0 2 0 . 1 1 4 5 0 8 0 0 0 . 0 0 1 5 7 6 8 6 0 . 0 0 1 5 2 1 1 7 - 0 . 0 0 0 0 1 3 1 1 - 0 . 0 0 0 0 2 9 2 9 0 . 0 0 0 0 1 7 5 6 - 0 . 0 0 0 0 1 1 6 4 0 . 0 0 0 0 3 6 4 5 - 0 . 0 0 0 0 5 0 4 7

1 . 6 0 . 1 5 1 1 1 4 9 5 0 . 1 5 1 1 4 7 5 1 0 . 0 0 3 6 0 4 6 2 0 . 0 0 3 6 3 9 5 4 0 . 0 0 0 0 2 3 2 9 0 . 0 0 0 0 0 6 2 6 0 . 0 0 0 0 1 0 5 2 - 0 . 0 0 0 0 0 8 3 2 - 0 . 0 0 0 0 0 8 0 2 - 0 . 0 0 0 0 2 9 9 1

1 . 8 0 . 1 9 7 7 8 1 7 5 0 . 1 9 7 7 8 6 1 6 0 . 0 0 8 6 3 6 2 4 0 . 0 0 8 6 1 0 8 0 0 . 0 0 0 0 6 9 1 2 0 . 0 0 0 0 5 5 9 7 - 0 . 0 0 0 0 3 7 9 4 0 . 0 0 0 0 0 6 1 4 0 . 0 0 0 0 0 1 1 2 - 0 . 0 0 0 0 1 3 4 3

1 . 9 0 . 2 2 6 0 7 6 3 0 0 . 2 2 6 0 5 3 9 5 0 . 0 1 3 4 3 8 3 4 0 . 0 1 3 4 1 3 0 9 0 . 0 0 0 2 4 1 7 6 0 . 0 0 0 1 9 3 1 4 0 . 0 0 0 0 2 6 7 6 - 0 . 0 0 0 0 0 2 9 9 0 . 0 0 0 0 1 6 3 1 - 0 . 0 0 0 0 0 3 3 3

2 . 0 0 . 2 5 8 9 6 1 6 1 0 . 2 5 8 9 5 1 6 9 0 . 0 2 1 3 3 2 7 6 0 . 0 2 1 3 8 7 6 7 0 . 0 0 0 5 8 0 0 8 0 . 0 0 0 5 8 7 2 6 0 . 0 0 0 0 0 5 0 2 - 0 . 0 0 0 0 0 4 9 0 0 . 0 0 0 0 2 2 4 7 - 0 . 0 0 0 0 2 8 1 8

2 . 1 0 . 2 9 7 5 8 0 3 7 0 . 2 9 7 5 8 0 5 4 0 . 0 3 4 9 1 8 9 9 0 . 0 3 4 9 6 3 3 9 0 . 0 0 1 8 2 0 8 3 0 . 0 0 1 8 2 7 8 2 0 . 0 0 0 0 0 9 4 9 0 . 0 0 0 0 2 4 4 2 0 . 0 0 0 0 2 0 7 3 0 . 0 0 0 0 1 4 7 8

2 . 2 0 . 3 4 0 9 6 4 3 8 0 . 3 4 0 9 4 2 8 6 0 . 0 5 6 9 2 8 4 2 0 . 0 5 6 9 0 6 4 4 0 . 0 0 5 4 5 4 2 5 0 . 0 0 5 4 5 5 7 8 0 . 0 0 0 3 0 4 8 2 0 . 0 0 0 3 2 8 3 1 - 0 . 0 0 0 0 1 0 4 3 - 0 . 0 0 0 0 2 8 4 9

2 . 3 0 . 3 8 5 7 0 7 6 6 0 . 3 8 5 6 4 8 8 2 0 . 0 8 8 4 5 1 4 8 0 . 0 8 8 4 2 4 2 4 0 . 0 1 4 3 6 1 4 5 0 . 0 1 4 3 3 0 5 5 0 . 0 0 1 6 8 2 2 4 0 . 0 0 1 7 4 4 5 8 0 . 0 0 0 1 9 4 0 3 0 . 0 0 0 1 1 7 0 1

2 . 4 0 . 4 2 5 1 6 5 2 8 0 . 4 2 5 1 7 1 9 6 0 . 1 2 3 8 4 2 9 7 0 . 1 2 3 8 9 3 7 7 0 . 0 2 9 5 5 8 2 9 0 . 0 2 9 5 7 9 7 9 0 . 0 0 6 0 5 6 0 1 0 . 0 0 6 0 1 4 6 0 0 . 0 0 1 0 7 4 7 5 0 . 0 0 1 0 4 9 9 0

2 . 5 0 . 4 5 6 6 6 6 5 1 0 . 4 5 6 6 7 1 4 6 0 . 1 5 5 3 8 2 9 3 0 . 1 5 5 4 1 4 3 8 0 . 0 4 6 8 2 2 1 0 0 . 0 4 6 7 5 9 5 8 0 . 0 1 2 9 3 0 6 9 0 . 0 1 2 9 2 8 4 5 0 . 0 0 3 3 4 9 7 2 0 . 0 0 3 2 9 5 8 4

2 . 6 0 . 4 8 2 2 9 8 5 4 0 . 4 8 2 2 7 6 9 3 0 . 1 8 2 1 0 0 1 3 0 . 1 8 2 0 7 6 0 1 0 . 0 6 3 0 0 8 1 7 0 . 0 6 3 0 4 1 2 3 0 . 0 2 0 7 2 5 5 4 0 . 0 2 0 6 9 1 8 3 0 . 0 0 6 5 1 7 3 0 0 . 0 0 6 4 2 3 3 4

2 . 7 0 . 5 0 4 6 3 2 4 2 0 . 5 0 4 6 2 3 4 5 0 . 2 0 6 3 4 6 0 1 0 . 2 0 6 3 7 2 2 7 0 . 0 7 9 0 0 6 0 4 0 . 0 7 9 0 7 1 6 2 0 . 0 2 9 2 1 7 2 4 0 . 0 2 9 2 6 2 0 1 0 . 0 1 0 5 2 3 7 7 0 . 0 1 0 5 1 1 1 4

2 . 8 0 . 5 2 4 3 7 4 0 0 0 . 5 2 4 3 6 2 3 2 0 . 2 2 8 2 4 2 1 9 0 . 2 2 8 2 7 4 4 0 0 . 0 9 4 0 8 5 3 6 0 . 0 9 4 1 5 4 3 4 0 . 0 3 7 7 7 2 6 8 0 . 0 3 7 7 2 7 7 9 0 . 0 1 4 8 1 6 9 8 0 . 0 1 4 7 8 2 7 2

Standard deviations for rectangular Wilson loops.

B e t a W i l s o n 2 x 1 W i l s o n 1 x 2 W i l s o n 3 x 2 W i l s o n 2 x 3 W i l s o n 4 x 3 W i l s o n 3 x 4 W i l s o n 5 x 4 W i l s o n 4 x 5 W i l s o n 6 x 5 W i l s o n 5 x 6

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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