Introduction to Lattice Gauge Theories

Introduction to lattice gauge theories
Rainer Sommer
DESY, Platanenallee 6, 15738 Zeuthen, Germany
WS 11/12: Di 9-11 NEW 15, 2’101
WS 11/12: Fr 15-17 NEW 15, 2’102
We give an introduction to lattice gauge theories with an emphasis on QCD. Requirements are quantum mechanics and for a better understanding relativistic quantum mechanics and continuum quantum ﬁeld theory.
These are not lecturenotes written to be easily readable (a script), but my private notes.
Still I am of course happy to receive corrections.
References
[1] J. Smit, Introduction to quantum ﬁelds on a lattice: A robust mate, Cambridge Lect. Notes
Phys. 15 (2002) 1–271.
[2] H. J. Rothe, Lattice gauge theories: An Introduction, World Sci. Lect. Notes Phys. 74
(2005) 1–605.
[3] I. Montvay and G. Mu¨nster, Quantum ﬁelds on a lattice, . Cambridge, UK: Univ. Pr.
(1994) 491 p. (Cambridge monographs on mathematical physics).
[4] M. Creutz, QUARKS, GLUONS AND LATTICES, . Cambridge, Uk: Univ. Pr. ( 1983)
169 P. ( Cambridge Monographs On Mathematical Physics).
[5] T. DeGrand and C. E. Detar, Lattice methods for quantum chromodynamics, . New Jersey,
USA: World Scientiﬁc (2006) 345 p.
[6] C. Gattringer and C. B. Lang, Quantum chromodynamics on the lattice, Lect.Notes Phys.
788 (2010) 1–211.
[7] C. Morningstar, The Monte Carlo method in quantum ﬁeld theory, hep-lat/0702020. Contents
1 Introduction 2
1.1 Particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Why a lattice? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Where will we go? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2Pathintegral in quantum mechanics 8
2.1 Euclidean Green functions in QM . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Quantum ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Problems with the path integral . . . . . . . . . . . . . . . . . . . . . . . 11
3Scalar ﬁelds on the lattice 13
3.1 Hypercubic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Green functions of the free ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Translation operator, spectral representation . . . . . . . . . . . . . . . . . . . 19
3.6 Timeslice correlation function and spectrum of the free thory . . . . . . . . . . 22
3.7 Lattice artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.8 Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.9 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4Gauge ﬁelds on the lattice 27
4.1 Color, parallel transport, gauge invariance . . . . . . . . . . . . . . . . . . . . . 27
4.2 Group integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 Some group integrals (for later) . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Pure gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.1 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.2 Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
11 Introduction
1.1 Particle physics
It is about the fundamental forms of matter (quarks = constituents of nucleons and their relatives) and their interactions. About scattering processes (LHC!) but also about the bound states: HOW is a nucleon made up of quarks.
The fundamental formulation is a quantum ﬁeld theory (or string theory, which for energies far below MPlanck is again a quantum ﬁeld theory). Field theories combine Poincare invariance and quantum mechanics.
1.2 Why a lattice?
Field theories have coupling “constants”, e.g. the ﬁne structure constant α in quantum electro dynamics (QED). The standard continuum treatment is an expansion in these coupling constants: “perturbation theory”.
QCD is the part of the theory which describes the by far dominant interactions of quarks
(up,down,charm,strange,top,bottom). It has a rather large eﬀective coupling constant at distances of the order (0.1 fm to 1 fm).
• There are phenomena, in particular in QCD, which are intrinsically non-perturbative: conﬁnement: quarks are never observed as free particles mproton = mp ꢀ 2mu + md
In fact in the limit of vanishing quark masses:
(1.1) mp ≈ mp|m =m =0 ∼ µ e−const./α (µ) = 0 to all orders in αs : 0 + 0α + 0α2 + . . . (1.2) sud
• the hadron mass spectrum is non-perturbative and can in principle be computed depending on just a few parameters (neglecting electromagnetism, weak interaction, gravitation):
αs, mu, md, ms, mc, mb, mt
• The lattice formulation is designed to enable such computations
• The lattice formulation is the only known formulation which contains perturbative and non-perturbative “sectors”.
It is therefore to be regarded as THE formulation (deﬁnition) of a quantum ﬁeld theory, in particular QCD.
1.3 Where will we go?
• Introduce the ingredients
- path integral, Euclidean time
- lattice (scalar ﬁelds)
- gauge ﬁelds (gluons)
- fermions (quarks)
2• Discuss concepts and computational methods
- continuum limit, Symanzik eﬀective theory
- strong coupling expansion
- MC method
- multiscale methods (maybe in part II)
• Discuss some results, e.g. quark — anti-quark potential
Figure 1: Static quark potential in the pure gauge theory. [2001]
3glueball spectrum pure gauge theory
12
0+−
3−−
2−−
1−−
2+−
3+−
1+−
10
2*−+
4
3
2
1
0
3++
0*−+
8
6
4
2
0
2−+
0−+
0*++
2++
0++
++ −+ +− −−
PC
Figure 2: Glueball spectrum in the pure gauge theory. [1998]
4hadron mass spectrum
1.4
1.2
1.0
0.8
K input
1.05
φ
K input
Ξ
1.00
0.95
Σ
Λ
K*
0.90
N
0.85
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2
a [GeV−1 ]
a [GeV−1 ]
K input
1.7
1.5
1.3
1.1
Ω
Ξ*
Σ*
∆
0.0 0.2 0.4 0.6 0.8 1.0 1.2
a [GeV−1 ]
Figure 3: Hadron mass spectrum. [2001]
52000
1500
1000
500
0
O
*
X
*
S
X
D
S
L
N
*
Krexperiment
Kwidth input p
QCD
Figure 4: Hadron mass spectrum. [2009]
6

chiral symmetry breaking decay rates: π → µν
[running of QCD coupling (maybe in part II)]
Figure 5: αs: left: lattice gauge theory [2005], right: experiment + PT [2004].
[interplay with eﬀective theories (maybe in part II) i.e. e.g. expansion in a or 1/mb]
[elastic scattering phases (maybe in part II)]
72Pathintegral in quantum mechanics
2.1 Euclidean Green functions in QM
We start from a QM Hamiltonian of one degree of freedom pˆ2
ˆ
H = + V (qˆ) (2.1) h/(2π) = 1, c = 1 .
2m
The statistical mechanics quantum partition function can be written as (T=1/temperature!)
Z
ˆ
Z = Tr e−TH =(2.2)
D[q] e−S[q]
q(0)=q(T)
(we will “show” that). The Euclidean action
Z
T
Y
.mq2
S[q] = (2.3) dt ( dq(t)“
+V (q)) , D[q] = “ 2
0
|{z }t
= L|t→it
A “derivation”:
We deﬁne the Euclidean time evolution operator (= transfer matrix) by a small (inﬁnitesimal) time unit a,
ˆˆ
−aH
T = e T(q, q0) = hq|e−aH|q0i ,
ˆ
,(2.4) such that
N
ˆ
Z = Tr T , N = T/a
(2.5)
Z
Z = (2.6) dq0 dq1 . . . dqN−1 T(q0, q1) T(q1, q2) . . . T(qN−1, q0)
Use the BCH formula (or explicitly) to show
1
2m pˆ2
T(q, q0) = hq|e−aV (qˆ)−a
|q0i
(2.7)
(2.8)
1
2m pˆ2
= hq|e−aV (qˆ)/2e−a e−aV (qˆ)/2|q0i + O(a3)
1
2
hq|e−aV (qˆ)/2|q0i = e−a V (q)δ(q − q0)
[(2.9) hp|p0i = δ(p − p0) ,
(2.10)
1hq|pi = e−ipq (2.11)
(2.12)
(2π)1/2
Z
11
2m pˆ2
2m p2 hq|e−a |q0i = dp hq|pihp|q0ie−a r
Z
112πm
11
0
2m p2
2a (q−q0)2
=dpe−ip(q−q )e−a =e−m (2.13)
2π 2π a](2.14) r
0
1q−q
(V (q)+V (q0))+ m ()2]
m
=e−a[ + O(a3) (2.15)
2a
2
2πa
And with
.qq(t + a) − q(t)
2
=(t + a/2) + O(a ) (2.16)
a
8we have naively shown the equivalence of quantum mechanics and a 1-dimensional Euclidean pathintegral. [remark about unboundedness, therefore a-expansion is not obvious]
Z
dq0 dq1 . . . dqN−1 T(q0, q1) T(q0, q1) . . . T(qN−1, q0)
Z=(2.17)
YY
[(2.18) dq(t) = q(ti) = D[q] ti
](2.19)
!
Z
N−1
Xaqi+1 − qi
const. × D[q] exp −
=(2.20) ()2 + V (qi) ,
qN = q0
2ai=0
Zconst. × D[q] exp(−S[q]) approximated on a 1-d lattice
=(2.21)
(2.22)
Za→0 const. × D[q] exp(−Scont[q]) .
−→
Euclidean Green functions: t1 ≥ t2
(2.23)
Z
1
G(t1, t2) (2.24) =
D[q] e−S[q] q(t1) q(t2) , ni = ti/a
Z
1
Z
=
dq0 dq1 . . . dqN−1 T(q0, q1) T(q1, q2) . . . T(qN−1, q0) qn qn (2.25)
12
Z
T/a −1 (T−t1)/a
(t1−t2)/a t2/a
ˆˆˆˆ
Tr
=)
( Tr T Tr T qˆT qˆT
(2.26)
=(2.27)
P
ˆˆ
n |nihn|e−E (T−t ) qˆe−H (t −t ) qˆe−H t n
1122
P
n T
Tr
n |nihn|e−E
P
ˆˆ
nhn|e−E (T−t ) qˆe−H (t −t ) qˆe−H t |ni n
1212
P
=(2.28)
n T
n e−E
[ the ground state is non-degenerate ] (2.29)
ˆˆˆˆ
T→∞ h0|eH t qˆe e(2.30) qˆe
|0i
−H t1 H t2 −H t2
1
−→
=(2.31) h0|qˆ(t1) qˆ(t2)|0i
ˆ ˆ qˆ(t) = eH tqˆe−H t = Euclidean Heisenberg operators (2.32)
And more generally: t1 ≥ t2 ≥ t3 . . . , T → ∞
G(t1, t2, ..., tn) = (2.34)
(2.33)
Z
1
D[q] e−S[q]q(t1) q(t2) . . . q(tn) .
Z
= h0|qˆ(t1) qˆ(t2) . . . qˆ(tn)|0i
(2.35)
(2.37) general ti , (2.36)
T → ∞
G(t1, t2, ..., tn) = h0|T {qˆ(t1) qˆ(t2) . . . qˆ(tn)} |0i
In lattice gauges theories, Euclidean Green functions = correlation functions = correlators are the central objects (but for 1+3=4 dimensions). They are mostly computed numerically by a Monte Carlo process. Assume one has the Euclidean Green functions. Real time physics is
9obtained (in principle) by analytic continuation (see later). But also directly:
(T → ∞)
ˆˆˆˆ
−H t1 H t2 −H t2
G(t1, t2) = G(t1 − t2) = h0|eH t qˆe |0i eqˆe (2.38)
(2.39)
(2.40)
1
ˆ
= h0|qˆe−H (t −t )qˆ|0ieE
0 (t1−t2)
12
X
n (t1−t2)
0 (t1−t2)
= h0|qˆ |nie−E hn| qˆ|0i eE n
X
=αn2 e−(E −E ) (t −t ) ,(2.41)
αn = hn|qˆ|0i
n
012
nfrom the large t2 − t1 behaviour: E1, E2, . . .. Improvement of precision by diﬀerent f(qˆ) .
In various places we dropped O(a2) terms. The result of a lattice path integral is only unique (universal) in the limit a → 0.
2.2 Quantum ﬁeld theory
Let us give a very rough overview what QFT (and therefore QCD) is about.
It describes
• particles moving in space and time |pi
• multiparticle states |p1, p2i: qq
H|p1, p2i ≈ ( p1 + m1 + p22 + m22)|p1, p2i
ˆ
(2.42)
22
≈ because there is always some (small) interaction
• scattering, decays, particle creation
• bound states → one of the main subjects of LGT .
Construction of QFT (scalar particles):
Creation and annihilation of particles at any point in space and time: QM operators at any point, quantum ﬁelds:
ˆqˆ → φ(x)
(2.43)
(2.44)
(2.46) ipˆ → πˆ(x) i
ˆˆ
[φ(x), φ(y)] = 0 = [πˆ(x), πˆ(y)] (2.45)
ˆ
[φ(x), πˆ(y)] = iδ(x − y) x not the QM variable, but more an index, labelling dof’s. Free particle:
Zm2
322
11
22
ˆ
ˆˆˆ
d x{ πˆ(x) + ∂jφ(x)∂jφ(x) + φ (x)}
H = (2.47)
2
Interactions: very restricted by general principles:
• unitarity, causality
10 • renormalizability → locality, dimension of ﬁelds in the Lagrangian
Only
λ0 λ0
Lint(x) = φ4(x) →
4
ˆ
Hint =φ (x) (2.48)
ˆ
44is possible. There is only one parameter: λ0, dimensionless.
Let us take two ﬁelds, combine them into one complex ﬁeld, corresponds in the end to a charged particle and its anti-particle.
1λ0
[φ1 + iφ2] , |φ| = (φ1 + φ2) , Lint(x) = |φ(x)|
2224
1
2
√
φ = (2.49)
4
2
The formal continuum path integral:
Z
Z = D[φ]D[φ†] e−S[φ] ,(2.50)
(2.52)
Z
λ0
24
S[φ] = (2.51) d4(x) {∂µφ∗(x)∂µφ(x) + m2|φ(x)| + |φ(x)| }
4
Z
G(x, y) ≡ hφ(x)φ∗(y) ≡ Z−1 D[φ]D[φ∗] e−S[φ]φ(x)φ∗(y) .
2.2.1 Problems with the path integral
It is a rather formal object:
• λ0 = 0: gaussian integrals as in QM
• λ0 0: perturbation theory (PT) in λ,
G(x, y) : (2.53)
G = G(0) + λ0G(1) + λ20G(2) + . . . (2.54) again gaussian integrals −→Feynman diagrams
• λ0 0: singular:
Z
∞
+|λ0|φ4(x) → ∞ e(2.55)
−∞
PT has a zero radius of convergence: asymptotic series
• Divergences: start with m0, λ0, compute
E2 = p2 + m2R (2.56)
(2.57) m2R = m20 + λ0 × (divergent integral) + O(λ20)) similar for λR
Reason: per unit volume: inﬁnite number of dof m0, λ0 → mR, λR involves inﬁnite (undeﬁned) relations
(2.58)
−→regularize, renormalize
11 • Regularize: modify short distances such that theory becomes deﬁned, in particular Feynman diagram integrals become ﬁnite, depend on some parameter. Inﬁnities appear when parameter is removed. e.g. a → 0. Most popular versions:
1 Regularize Feynman diagram integrals,
RRe.g. “dimensionally” d4p → dnp , n = 4 − ꢀ
2 Regularize the path integral: lattice with spacing a • Renormalize: take a limit of some regularization parameter (ꢀ → 0 , a → 0) at ﬁxed mR, λR observable(mR, λR) = lim {observable(a, m0, λ0)}
(2.59)
(2.60) mR,λR a→0 m0(a) , λ0(a) , : lim λ0(a) = ∞ is possible (allowed) a→0 more precisely, everything is dimensionless, measure masses etc in units of a observable(mR, λR) = lim {observable(am0, λ0)}
(2.61) mR,λR amR→0 am0(amr) , λ0(amr) , : lim λ0(amr) = ∞ is possible (allowed) (2.62) amr→0 bare parameters = parameters in the Lagrangian are not observable, irrelevant; once they are eliminated there are no divergences.
1) Dimensional regularization mixes deﬁnition and and approximation gives asymptotic expansion in λR is tremendously successful (QED!)
2) Lattice regularization non-perturbative deﬁnition of the theory validity, precision of expansion in λR can be checked
12 3Scalar ﬁelds on the lattice
3.1 Hypercubic lattice lattice:
Λ = aZ4 = {xµ = anµ| nµ ∈ Z , µ = 0, 1, 2, 3}
(3.1)
ﬁnite lattice = lattice in ﬁnite volume
Λ = {xµ = anµ| nµ = 0, 1, . . . Lµ/a}
(3.2)
mostly
L0 = T , L1 = L2 = L3 = L . (3.3)
Having the same spacing in all directions enhances the symmetry and is relevant.
Integral as for the action
Z
XX
S = d4x L(x) → a4
L(x) ≡ a4
L(an0, an1, an2, an3) with xµ = anµ . (3.4) xn0,n1,n2,n3
Derivatives:
1∂µφ(x) ≡ (∂µφ)(x) = [φ(x + aµˆ) − φ(x)] forward: (3.5) backward: (3.6) symmetric: (3.7) a
1
∂µ∗φ(x) ≡ (∂µ∗φ)(x) = [φ(x) − φ(x − aµˆ)] a
11
∗
˜
∂µφ(x) ≡ (∂µ + ∂µ)φ(x) = [φ(x + aµˆ) − φ(x − aµˆ)]
22a
Partial integration
ZZd4x (∂µφ∗(x))(∂µφ(x)) = − d4x φ∗(x)∂µ∂µφ(x) (3.8)
is valid
– in ﬁnite volume with pbc: φ(x + Lµµˆ) = φ(x)
– in inﬁnite space-time and ﬁelds which vanish fast enough at inﬁnity
Partial summation
XXa4 ψ(x)∂µφ(x) = a3 (3.9)
ψ(x)[φ(x + aµˆ) − φ(x)]
xx
XX
= a3[ (3.10) ψ(y)φ(y)]
ψ(y − aµˆ)φ(y) −
yy
X
= −a4
(∂µ∗ψ(y)) φ(y) (3.11) y
[ ψ(x) = ∂µφ∗(x) ] (3.12)
XXa4 (3.13)
∂µφ∗(x)∂µφ(x) = −a4
φ∗(x)∂µ∗∂µφ(x) . xx
13 This gives the action for a scalar ﬁeld on the lattice
X
λ0
24
S = a4 (3.14)
{∂µφ∗(x)∂µφ(x) + m2|φ| (x) + |φ| (x)}
4x
X
λ0
= a4 (3.15)
{−φ∗(x)∂µ∗∂µφ(x) + m2|φ| (x) + |φ| (x)}
24
4x
= φ∗nMnmφm = φ†Mφ , φn = aφ(na) , n = (n0, n1, n2, n3) , (3.16)
M = M† , M 0 . (3.17)
Check as an exercise
Ordered the ﬁelds in some way ... Let us study the free ﬁeld, λ0 = 0. We expect particles, states with a deﬁnite momentum. Decoupled (free). So expect that the action is diagonalized in momentum space:
˜† ˜ ˜
S = φ Mφ , Mnm = µ˜2mδmn
˜
.(3.18)
Let’s look at momentum space ﬁrst.
3.2 Momentum space
Plane waves are eipx = eip x .(3.19)
µµ
Expanding a ﬁeld in plane waves is a Fourrier transformation,
Zd4p ipx
˜f(x) = (3.20) e f(p) .
(2π)4
Since eip x = ei(p x +2πn ) = ei(p +2π/a)x (3.21)
(3.22)
µµµµµµµthe momenta
2π pµ ⇔ pµ + aare equivalent, and we can restrict them to the Brillouin zone,
ππaa
−≤ pµ
.(3.23)
So we have
Z
π/a d4p ipx
˜f(x) = (3.24) e f(p)
(2π)4
−π/a in inﬁnite volume.
Finite volume
If in addition we put ourselves into a ﬁnite volume, there are clearly also a ﬁnite number
Qof momenta, µ Lµ/a of them. We can then write a ﬁeld as
X
1ipx
˜f(x) = e f(p) , V = L0L1L2L3 = T L1L2L3 . (3.25)
Vp
14 Usually one chooses periodic boundary conditions (PBC),
XXf(x + µˆLµ) = eiθ f(x) → eipxeip L f(p) = (3.26) e e f(p) ,
ipx iθµ
µµµ
˜˜pp
(no summation over µ) and one has
2π θµ eip L = eiθ → pµ = kµ + (3.27) ,
kµ ∈ {0, 1, . . . , Lµ/a − 1}
µµµ
Lµ Lµ
2a
Lµ Lµ
(or a diﬀerent range: − ≤ pµ 2a ). Normally one has
θµ = 0 “PBC” (3.28)
θµ = π (3.29) “APBC”
but general θµ allows more ﬂexibility.
As the volume becomes bigger, our spacings in momentum space
2π
ꢀµ = (3.30)
Lµ shrink and XX
11ipx ipx
˜˜f(x) = e f(p) = ꢀ0 . . . ꢀ3 e f(p) (3.31) V(2π)4 pp
Zd4p ipx
˜
→e f(p) . (3.32)
(2π)4
15 In the exercise we show a4
XYeip(x−y) =δn m (3.33)
[ xµ = anµ , yµ = amµ ]
µµ
Vpµ
|{z }
=a4 δ(x−y)
P
δ(x − y): lattice delta-function: a4 x δ(x − y) f(x) = f(y) and we always identify f(x) = f(x + aµˆLµ).
We can take the inﬁnite volume limit
Z
π/a d4p
(2π)4 eip(x−y) = δ(x − y) .
(3.34)
−π/a
Of course we also have (written in a diﬀerent way): a4 (2π)4
XYYeix(p−q) (3.35) =
δn m = δ(p − q) ꢀµ = δ(p − q)
µµ
VVxµµ
[ pµ = nµ2π/Lµ , qµ = mµ2π/Lµ ] (3.36)
Xxa4 (3.37) eix(p−q) = (2π)4 δ(p − q)
3.3 Green functions of the free ﬁeld
X
S = φ∗nMnmφm = φ†Mφ , φn = aφ(na) . (3.38)
mn
M = M† , M 0 , (3.39) so there is a unitary matrix U,
U†U = 1 , (U†MU)nm = δmnµ2m , µm ∈ R.
(3.40)
¯
To solve the free theory, we introduce a generating functional (J, J = sources)
Z
P
4
∗
¯
x[J(x)φ(x)+φ (x)J(x)]
¯
Z(J, J) = D[φ]D[φ∗] e−S[φ]+a ,(3.41)
ꢀ
ꢀ
∂∂hφ(x)φ∗(y)i = Z(0, 0)−1a−8
Z(J, J) (3.42) .
¯ ꢀ
ꢀ
¯
∂J(y)
∂J(x)
¯
J=J=0
It is evaluated as
Z
P
˜
m[−|φ|mµ2m+Jmφm+φmJm †
]
2
∗
¯˜
˜˜˜
¯
˜
Z(J, J) = D[φ]D[φ∗] e
[ φm = Umnφn = Un∗mφn , . . . (3.43)
∗∗†∗
˜˜
D[φ]D[φ ] = D[φ]D[φ ] det U det U = D[φ]D[φ ] ] (3.44)
Z
Y
˜
2
∗
¯˜
˜˜˜
∗−|φm| µ2m+Jmφm+φmJm
˜˜
=dφmdφm e (3.45)
m
Y
mJmµ−m2
T M−1J
π
˜
¯˜¯
=eJ = (det(M/π))−1 eJ (3.46)
µm2 m
P
8
¯
= (det(M/π))−1 ea (3.47)
x,y J(x)G(x,y)J(y)
16 with with
−1
Jn = a3J(na), M(3.48)
= a2G(an, am)
(3.49) nm
[−∂µ∗∂µ + m2]G(x, y) = δ(x − y)
From this we get the two-point function
∂∂
Z(0, 0)−1 Z(J, J) (3.50)
¯
¯
∂J(y)
∂J(x)
X
P
∂
8
0
¯
J(z)G(z,z )J(z)
0z,z
=(3.51) a8[
G(x, z)J(z) ] × ea
∂J(y) z
P
8
0
¯
J(z)G(z,z )J(z)
= a8G(x, y) ea
(3.52)
0z,z
¯
+ terms that vanish when we set J = J = 0 (3.53)
→ hφ(x)φ∗(y)i = G(x, y) .
(3.54)
The matrix U which diagonalizes M is the matrix formed from the fourrier transformation.
This is due to translation invariance (periodic boundary conditions or inﬁnite volume) and the fact that M just contains ﬁnite diﬀerences as non-trivial terms. So we have
X
1
˜
G(x, y) = G(x − y, 0) = eip(x−y) G(p) a graph (3.55)
Vp
11
G(p) = + O(a2 p2µ) exercise (3.56)
˜
∼pˆ2 + m2 p2 + m2
Exercise
What are the higher point functions, such as G(u, v, w), G(u, v, w, x)?
3.4 Transfer matrix
We treat here a real scalar ﬁeld. The complex one is basically two copies of the real one.
The simple scalar action allows for the explicit derivation of a transfer matrix just like in quantum mechanics.
We set a = 1 (all dimensionful quantities in units of a until eq. (3.79)). The action is m2 λ0
X
φ2(x) + φ4(x)}
1
2
S = (3.57)
{ ∂µφ(x)∂µφ(x) +
24x
The transfer matrix acts between timeslices. Therefore we collect all variables in a timeslice:
Φ(x0) = {φ(x0, x) , x ∈ Λspace }
(3.58)
17 The action is rewritten as
X X
1 [φ(x0 + 1, x) − φ(x0, x)]2 + V (Φ(x0))
S = (3.59)
2x0 xm2 λ0
X
φ2(x) + φ4(x)} .
1
2
V (Φ(x0)) = (3.60)
{ ∂jφ(x)∂jφ(x) +
24x
We want to show as in QM
Z
ˆ
Z = Tr e−TH =.(3.61)
D[φ] e−S
ˆ
First we need a Hilbert space and operators. Introduce φ(x) as operator φ(x) and canonical conjugate πˆ(x) (Schro¨dinger picture):
ˆˆ
[φ(x), φ(y)] = 0 = [πˆ(x), πˆ(y)] (3.62)
Y
ˆ
[φ(x), πˆ(y)] = iδ(x − y) ≡
δx y (3.63) i i iand just as usually (think of φ(x) = q, π(x) = p) hφ0(x)|φ(x)i = δ(φ0(x) − φ(x)) hπ0(x)|π(x)i = δ(π0(x) − π(x))
(3.64)
(3.65)
(3.66)
00
ˆhφ (x)|F(φ(x))|φ(x)i = δ(φ (x) − φ(x))F(φ(x))
1hφ(x)|π(x)i = e−iφ(x)π(x) .(3.67)
(2π)1/2
The total Hilbert space is the direct product of Hilbert spaces at each x
Y
|Φi = |φ(x)i = |φ((0, 0, 0))i|φ(a, 0, 0)i . . . |φ((L − 1, L − 1, L − 1))i .
(3.68) x
An explicit representation is the Schro¨dinger representation:
Z
Y
2
ψ[u] = ψ[{u(x)}] , hψ|ψi = du(x) |ψ[u]| (3.69) x
ˆ
φ(x) ψ[u] = u(x) ψ[u] (3.70)
∂
πˆ(x) ψ[u] = −i
∂u(x)
ψ[u] . (3.71)
We factorize the “Boltzmann factor”
Y
P
111
222
x[φ(x0+1,x)−φ(x0,x)]2
V (Φ(x0)) e−S =e− V (Φ(x +1)) e− e− (3.72) 0x0
Now consider the diPﬃcult term (as in quantum mechanics):
Y
11
22
x[φ(x0+1,x)−φ(x0,x)]2
[φ(x0+1,x)−φ(x0,x)]2 e− =e− (3.73)
=(3.74)
(3.75) x
Y
1
2
πˆ(x)2 hφ(x0 + 1, xP)|e−
|φ(x0, x)i x
1
2
x πˆ(x)2
= hΦ(x0 + 1)|e− |Φ(x0)i
18 and then
The TM,
P
111
222
x[φ(x0+1,x)−φ(x0,x)]2
V (Φ(x0)) e− V (Φ(x +1)) e− e− (3.76) 0
P
111
222
x πˆ(x)2 V (Φ)
ˆˆ
= hΦ(x0 + 1)|e− V (Φ) e− e− (3.77)
|Φ(x0)i .
P
111
222
ˆˆ
−
x πˆ(x)2 V (Φ)
T = e V (Φ) e− e− .(3.78)
(3.80)
ˆis hermitian and positive. Therefore (restoring a) writing
ˆ
−aH
ˆ
T = e
,(3.79)
ˆmakes sense. It deﬁnes the lattice hamiltonian H, a hermitian operator.
ˆˆ† ˆ
H = H , H ≥ 0 .
A formal expansion in a gives: m2 λ0
X
2422
11
22
ˆ
ˆˆ
ˆˆ
{ πˆ(x) + ∂jφ(x)∂jφ(x) +
φ (x) + φ (x)} + O(a ) .
H = (3.81)
24x
Remarks:
• The exisitence of a positive, hermitian operator is called positivity. It corresponds to unitarity in Minkowsky space, the conservation of probablity, very important.
ˆ
• An action with just 2nd order derivatives was important in deriving T.
• We emphasize the huge Hilbert space. A QM degree of freedom at each space-point.
3.5 Translation operator, spectral representation
Let us introduce the spatial translation operator by
0
ˆ
U(x)|ψi = |ψ i ,
(3.82)
(3.83)
00
ˆˆhψ1|φ(y)|ψ2i = hψ1|φ(y − x)|ψ2i .
This means that
† ˆ ˆ
ˆˆ
hψ1|U(x) φ(y)U(x)|ψ2i = hψ1|φ(y − x)|ψ2i
(3.84)
(3.85) and therefore the operators transform as
† ˆ ˆ
ˆˆ
U(x) φ(y)U(x) = φ(y − x) .
Clearly our Hamiltonian is chosen invariant
ˆ† ˆ ˆ ˆ
U(x) HU(x) = H . (3.86)
19 ˆ
Another property that we need is that U(x) is unitary. We look at the Schro¨dinger representation
Z
Z
Z
Y
ˆhψ1|φ(y − x)|ψ2i = dv(z) ψ1∗[v]v(y − x)ψ2[v]
(3.87)
z
Y
=(3.88) dv0(z) ψ1∗[v0] v0(y) ψ2[v0] , [ v0(y) = v(y + x) ] z
Y
0∗000
ˆ
=dv(z) (ψ1) [v] v(y) ψ2[v] = hψ1|φ(y)|ψ2i
(3.89)
z
000
ˆ
ψi[v] = U(x)ψi[v] = ψi[v ] , v (y) = v(y + x) (3.90)
From this we see immediately that
00
ˆ† ˆ hψ1|ψ2i = hψ1|U(x) U(x)|ψ2i = hψ1|ψ2i .
(3.91)
ˆˆˆ
There are simultaneous eigenstates of H and U. For eigenstates of U we have
2
U(x)|λ(x)i = λ(x)|λ(x)i , |λ(x)| = 1 , λ(x) = eiα(x)
(3.92)
(3.93)
ˆ
α(x) + α(y) = α(x + y) → α(x) = −xp with some p .
The last equation is just because of the linearity seen before. So we have with |λ(x)i = |p, ni, n for other quantum numbers,
−ixp
ˆ
U(x)|p, ni = e |p, ni ,
(3.94)
(3.95)
ˆ
H|p, ni = (E(p, n) + E0)|p, ni ,
E0 (3.96) :
the ground state energy
The physical interpretation is that p is the momentum of the state. And on a ﬁnite, periodic lattice we have the restrictions for p as before. As normalization we choose hp, n|p0, n0i = 2 E(p, n) L3 δ(p − p0) δnn
(3.97)
0
Let us now look at the 2-point function and use translation invariance to derive the spectral representation.
T → ∞
,L ﬁnite (3.98)
(3.99)
G(x − y) = hφ(x) φ(y)i
ˆ
E0 (x0−y0)
ˆ−H |x −y | ˆ
00
=
= h0|φ(x)e φ(y)|0i e
(3.100)
X
11e−E(p,n) |x −y |h0|φ(x)|p, nihp, n|φ(y)|0i
00
ˆˆ
L3 2 E(p, n) p,n
ˆ† and, assuming the translation invariance of the ground state, U (x)|0i = |0i (for a ﬁnite system it can be proven that the ground state is translation invariant)
ˆˆ
ˆˆ† h0|φ(x)|p, ni = h0|U(x)φ(0)U (x)|p, ni
(3.101)
(3.102)
= eipxh0|φ(0)|p, ni
ˆip(x−y)
2
ˆˆˆh0|φ(y)|p, nihp, n|φ(x)|0i = e |h0|φ(0)|p, ni|
(3.103)
20 This gives
2
X
ˆ
1
|h0|φ(0)|p, ni|
G(x − y) = eE(p,n) |x −y | eip(x−y) (3.104)
00
L3 2E(p, n) p,n
Z
X
1
L3 dω =ρL(ω, p) e−ω |x −y | eip(x−y) (3.105) ,
00
p
2
X
ˆ
|h0|φ(0)|p, ni|
2E(p, n)
ρL(ω, p) = (3.106)
δ(ω − (E(p, n))) .
n
And in inﬁnite volume
G(x − y) →
ZZd3p
(2π)3 dω ρ(ω, p) e−ω |x −y | eip(x−y) ,(3.107)
00
ρ(ω, p) = lim ρL(ω, p) = spectral density. (3.108)
L→∞
From our derivation we have seen that the spectral density is nothing but the coupling of a 2
ˆ
ﬁeld-operator to states of deﬁnite momentum: |h0|φ(0)|p, ni| . The x-dependence of the twopoint function follows from space and time translations, in terms of two dynamical quantities,
E(p, n) , ρ(ω, p).
21 3.6 Timeslice correlation function and spectrum of the free thory
The free propagator is
ZZd4p d3p
(2π)4 (2π)3 ipx e G(p) = eipxG(x0; p) (3.109)
˜
G(x) = hφ(x)φ(0)i = Z
π/a dp0 e−ip x G(p) . (3.110)
00
G(x0; p) =
(2π)
−π/a
From the above use of translation invariance we have
Z
Xdω ρ(ω, p) e−ω|x | =.(3.111)
c2n(p) e−E(p,n)|x |
00
G(x0; p) = n
We now evaluate this explicitly.
Z
π/a dp0 eip x
(2π) pˆ2 + m2
00
G(x0; p) = (3.112) for x0 ≤ 0
−π/a
[ φ = ap˜0 , n0 = −x0/a ]
(3.113)
Z
πdφ 1
= a eiφn (3.114)
0
2 ˆ2
22
(2π) a m + a p + 2(1 − cos φ)
−π
Z
πdφ 1
2 ˆ2
22= a eiφn [ A = 2 + a m + a p ] (3.115)
0
(2π)
A − 2 cos φ
−π
[ z = eiφ , dz = izdφ , 2 cos φ = (z + z−1) ] (3.116)
I
X
11
= a dz zn (3.117) =Residues
0
2πi z[A − (z + z−1)]
|z|=1
(3.118)
The poles are at (ω 0)
D = z[A − (z + z−1)] = 0 , z1 = e−aω , z2 = eaω
A = 2 cosh(aω) ω 0 , (3.120)
→ D = −(z − e−aω)(z − eaω) .
,(3.119)
(3.121)
Only z1 is inside the circle. Its residue is e−n aω
0
.(3.122)
2 sinh(aω)
So we have e−|x |ω
0
G(x0; p) = ,(3.123)
(also for x0 0)
2 sinh(aω)/a
22
2 ˆ2
→ E(p) = ω(p) , 2[cosh(aE(p)) − 1] = a m + a p ,
(3.124)
1
ρ(ω, p) = aδ(ω − E(p))
2 sinh(aω) spectral density: (3.125)
We observe
22 • There is only one intermediate state per p; the spectral density is a single δ-function.
Such a state is created from the vacuum by
X
3ipx
ˆ
ˆ
O1(p) = Ca φ(x)e .(3.126)
x
Namely we had
ˆ
G(x0 − y0; p) ∝ hO1(x0, −p)O1(y0, p)i ∝ h0|O1(p)†e−|x −y |HO1(p)|0i
(3.127)
00
• The energy momentum relation is (when a2 pi2 ꢁ 1 , a2m2 ꢁ 1):
E2 = p2 + m2 + O(a2) (3.128)
⇒ a free particle with relativistic energy momentum relation.
• 2-particle states are simply created by
ˆˆˆˆˆ
O2(p, q) = O1(p) O1(q) = O1(q) O1(p) (3.129)
A correlation function is
X
G4 = a12 (3.130) e−i(pz+qw−px−qy)hφ(t, z) φ(t, w)φ(0, x)φ(0, y)i x,y,z,w this gives: a graph time dependence C1(p, q) e−|t|(E(p)+E(q)) + C2(p, q) δ(p + q) (3.131)
So
E = E(p) + E(q) ≥ 2m a graph (3.132)
For later use we note that above we have shown that
Z2π dp0 sinh(aω) cosh(aω) − cos(ap0) e−|x |ω =.(3.133)
eip x
00 0
3.7 Lattice artifacts
Let us include the O(a2) eﬀects:
22
[cosh(aE(p)) − 1] = E2 + a2E4 + O(a4)
(3.134)
a2 4!
(m2 + p2)2a2
= E2 + (3.135)
+ O(a4)
12 p4j a2
2pˆ2j =(3.136)
(1 − cos pja) = p2j −
+ O(a4) ,
a2 12 a2
X
E2(p) = m2 + p2 − [(m2 + p2)2 + p4j ] +O(a4) . (3.137)
12 j
|{z }
≈10% when aE(p)=1
23 Now we have to be careful, however. This includes the mass in the Lagrangian, not an observable. We should renormalize ﬁrst, even at tree level! Not unique, but very natural: mass = energy at rest : renormalization condition (3.138)
(3.139) a2 m2R = E2(p = 0) = m2 (1 − m2 + . . .)
12 a2 m2 = m2R (1 + m2R) + O(a4) (3.140)
(3.141)
12
→a2
X
E2(p) = mR2 + p2 − [2mR2 p2 + (p2)2 + p4j ] +O(a4) . (3.142)
12 j
|{z }
≈10% when aE(p)=1
See Fig. 5, left.
Can this be improved? Better discretization?
3.8 Improvement
In general there is Symanzik improvement:
X X
S → Simpr = S + δS , δS = a4 ciad −4Oi(x) , (3.143)
Oxi
Here we have
3a2
XXciad −4Oi(x) = c1
[∂µ∂µφ(x)]2 (3.144)
O
2i
µ=0
3
X
= [ ∂µ∂µφ(x) ]2 (3.145)
µ=0
X
22 2
1
→E2(p) = m2 + p2 + a2[c − 12 ][(m + p ) + p4j ] + O(a4) (3.146) j
Remarks (here not really explained):
• In general in a scalar theory in 4 dimensions: Oi(x) local ﬁelds with mass dimension 6
Even more generally: local ﬁelds with mass dimension ≥ 5 ci = ci(λ0) (3.147)
• On-shell improvement: terms which vanish by the e.o.m can be dropped (see exercise for the e.o.m.)
• Also ﬁelds in correlation functions have to be improved (correction terms). We here looked only at energies (which are on-shell).
24 Figure 6: E2(p)/E2(p)cont for p = (p, 0, 0), m = 0, against the square lattice spacing a2E2. Left side: c = 0, right side: c = 1/12.
3.9 Universality
Z
Z = D[φ] e−S[φ] ,(3.148)
(3.149)
Zhφ(x)φ(y)i = Z−1 D[φ] e−S[φ]φ(x)φ(y) .
Xx0→∞
Rx0 hφ(x)φ(0)i a3 e−E(p=0)x = e−m (3.150)
= e−n /ξ
00
∼x
A change of notation
S → H/(kTtemp
)(3.151)
(3.152)
(3.153)
(3.154) d = 3 + 1 → d = 4 (+0 : static)
(amR)−1 → ξ a → 0 → ξ → ∞ shows that a lattice ﬁeld theory in 3 + 1 dimensions is a statistical model in 4 dimensions. The continuum limit is reached at a critical point. Statistical models are known to have universality there.
Universality means that a change of the Hamilton function which does not change the symmetries (axis permutations, φ → −φ etc.) gives the same correlation functions.
In the QFT this means the continuum limit is unique; it does not depend on the details of the discretization (adding e.g. (∂µ∂µφ)2 term).
More precisely: the continuum limit only depends on the coeﬃcients of the renormalizable interaction terms: [O(φ(x), ∂µφ(x)] ≤ 4 . Eg. the addition of φ6 changes only the cutoﬀ eﬀects.
In terms of continuum ﬁeld theory best looked at in the corresponding action: m2
X
S = a4 (3.155)
{ ∂µφ(x)∂µφ(x) +
φ2(x)} + Sint
1
2
2x
25 Figure 7: E2(p)/E2(p)cont against the square lattice spacing a2E2. Improved case: c = 1/12. Left side: for p = (p, p, 0), m = 0, right side: p = (p, 0, 0), mR = p. then
P
• assume locality: Sint = a4 x const. M−(n+4k−4) φn(x)(∂µφ(x)∂µφ(x))k + . . .
• renormalizability when (we just state this):
X
Sint = a4 (3.156)
Lint(x) , [Lint(x)] ≤ 4
x
.[Φ] = mass dimension: [∂µ] = 1 , [φ(x)] = 1 , (3.157) and no odd powers of φ and Euclidean invariance
λ0
λ0
4
Lint(x) = φ4(x) →
ˆ
Hint =φ (x) (3.158)
ˆ
44
λ0 is the only possible coupling constant, free parameter
QFT’s are very predictive
• how well is this established (proven)? to all order in λ0 → λR (T. Reisz, lattice power counting theorem) non-renormalizability of eg. φ6(x)/m62 in the same way to all orders in m6−2 .non-perturbatively: numerical investigations are in agreement with this.
Remark: do not confuse higher dimensional operators in Symanzik improvement and a theory with higher dimensional operators in the continuum limt. In one ca−se4 the coeﬃcients are proportional to ad −4, in the other case they are proportional to 1/md .
OO
26 4Gauge ﬁelds on the lattice
4.1 Color, parallel transport, gauge invariance
Quarks carry color, A, B = 1 . . . 3 (1 . . . N in an SU(N) gauge theory):

ψ1(x)
3

ψ(x) = ψ2(x) ∈ C .
ψ3(x)
(4.1)
The basic principle is gauge invariance: diﬀerent colors are completely equivalent. One can rotate the ﬁelds and physics does not change.
Λ(x) ∈ SU(3) : ψ(x) → ψΛ(x) = Λ(x) ψ(x) .
(4.2)
(There is also the possibility to rotate by a phase ( U(1) ), but this corresponds to electrodynamics, a separate issue). Λ(x) ∈ SU(3) is an arbitrary function of x. So it makes no sense to compare ψ(x) and ψ(y). Before comparing we have to parallel-transport ψ(y), such that it transforms as ψ(x). Parallel transporter:
P(x ← y) : P(x ← y) → PΛ(x ← y) = Λ(x) P(x ← y) Λ−1(y)
P(x ← y) ψ(y) → Λ(x) P(x ← y) ψ(y) .
(4.3)
(4.4)
The parallel transporter will in general depend on the path from y to x. Think of a straight path for deﬁniteness. For the deﬁnition of a (continuum derivative) we need the transporter by an inﬁnitessimal distance:
Dµψ(x) = (Dµψ)(x) = lim[P(x ← x + ꢀµˆ) ψ(x + ꢀµˆ) − ψ(x)] .
(4.5)
ꢀ→0
Every SU(N) matrix can be written in the form
N2−1
X
P = eB , B = −B† =
BaTa (4.6)
(4.7) a=1
Ba ∈ R , T = −(T ) , tr T = 0 , tr T T = − δab , [T , T ] = fabcTc . a † aaabab
1
2
For example in SU(2):
1
Ta = τa , Pauli matrices . (4.8)
2i
So we can write for an inﬁnitessimal path
P(x ← x + ꢀµˆ) = eꢀA (x) →
Dµ = ∂µ + Aµ . (4.9)
µ
Aµ has mass dimension one. If quarks were scalars, gauge invariance would force the kinetic term in the Lagrangian to look like
X
2
|DµψA| .
(4.10)
A
27 It contains a coupling to the ﬁeld Aµ (gluon ﬁeld), an interaction term. What about a kinetic term for Aµ? Gauge invariance is the basic principle. First in the continuum:
The basic object is Dµ because it is gauge covariant,
DµΛ = Λ(x)DµΛ−1(x) . (4.11)
We want something Euclidean invariant and gauge invariant: tr DµDµ = tr (∂µ∂µ + . . .) ∂µ acting on what? (4.12)
(4.14)
(4.15)
tr Fµν Fµν , (4.13)
Fµν = [Dµ, Dν] = ∂µAν − ∂νAµ + [Aµ, Aν]
= Ta{∂µAaν − ∂νAµa + fabcAµb Aνc }
= TaFµaν . (4.16)
Other terms have higher dimension, are not renormalizable. So the action is
ZZ
1
1
2g02
SG = − d4x tr Fµν(x)Fµν(x) = (4.17) d4x F2 .
2g02
|{z} convention
Unlike QED, there are interaction terms already in here even without the quark ﬁelds.
On the lattice, the quark ﬁelds are at the lattice points x, but the gluon ﬁeld has to provide the parallel transporter from point to point, it is sitting on a link
P(x ← x + aµˆ) = U(x, µ)
P(x + aµˆ ← x) = U−1(x, µ)
:(4.18)
.:(4.19)
Gauge covariant derivative:
1Dµψ(x) = [U(x, µ)ψ(x + aµˆ) − ψ(x)] forward: (4.20)
a
1backward: (4.21)
Dµ∗ψ(x) = [ψ(x) − U−1(x − aµˆ, µ)ψ(x − aµˆ)] a
With these we could build a kinetic term for scalar quarks
X X X X X
2
Sφ = a4 (4.22)
|DµψA| = a4
ψA∗ (−Dµ∗Dµψ)A = a4
ψ†(−Dµ∗Dµ)ψ .
xxx
AA
For fermions we do that later.
For the kinetic term we need a local object, gauge invariant. The most local one is the parallel-transporter around a plaquette (elementary square): x + a ν
Oµν(x) = tr U(x, µ)U(x + aµˆ, ν)U−1(x + aνˆ, µ)U−1(x, ν) : (4.23)
.
xx + a µ
28 Assume that we have a smooth classical ﬁeld, so the parallel transporters are close to one, we can then write
U(x, µ) = eaA (x) (4.24)
Then we can expand in a (classical continuum limit)
Oµν(x) = tr U(x, µ)U(x + aµˆ, ν)U−1(x + aνˆ, µ)U−1(x, ν) (4.25)
µ
= tr eaA (x)eaA (x+aµˆ) (4.26) e,−aA (x+aνˆ)e−aA (x)
µνµν
1
|{z } | {z }eB eC
23
B = a(Aµ(x) + Aν(x + aµˆ)) + a [Aµ(x) , Aν(x + aµˆ)] + O(a ) , (4.27)
(4.28)
2
23
1
C = a(−Aµ(x + aνˆ) − Aν(x) + a [Aµ(x + aνˆ) , Aν(x)] + O(a ) ,
2
1
2
[B,C]+... eBeC = eB+C+ (4.29)
= eD ,
(4.30)
D = −a2∂νAµ(x) + a2∂µAν(x) + a2[Aµ(x) , Aν(x)] + O(a3)
= a2 Fµν(x) + O(a3) (4.31)
We note that
D = DaTa , (4.32)
→tr D = 0 .
This is true for the O(a2) terms but also for the higher ones: they are formed from derivatives or from commutators. Commutators always can be written again as CaTa . Therefore we can write
D25
1
2
Oµν(x) = tr e = N + tr D + tr D + O(a ) (4.33)
425
1
= N + a tr (Fµν(x)) + O(a ) (4.34)
2x + a νxx + a µ
= tr U(p) , p = (x, µ, ν) (4.35)
And ﬁnally have the Wilson plaquette action
.
XX X
g02
11g02
SG[U] = (4.36) tr {1 − U(p)} = tr P(x, µ, ν) ,
pxµ,ν
P(x, µ, ν) = 1 − U(x, µ) U(x + aµˆ, ν) U(x + aνˆ, µ)−1 U(x, ν)−1
.(4.37)
Of course other forms are possible. In fact, take any small loop on the lattice (a graph), sum over all orientations. The result will be
XX
4tr (Fµν(x))2 + O(a2) , (4.38)
˜
Oµν(x) = c1 + c2a x,µ,ν x,µ,ν because there is no other gauge invariant ﬁeld of dimension ≤ 4. It was therefore not really necessary to do the above calculation. By the same logics, there is no dimension ﬁve axispermutation invariant ﬁeld. This is why the next term is dimension 6, giving O(a2) for the lattice spacing corrections.
29 4.2 Group integration
We remain in the pure gauge theory. In order to fully deﬁne the path integral we have to specify the integration measure. The variables are in the group SU(3), so we want to know dU =? , U ∈ SU(N) .
The basic principle is gauge invariance, so we want the measure to be gauge invariant.
Parametrize the SU(N) matrices by
(4.39)
W(ω) = exp(ωaTa) . (4.40)
A gauge transformation gives
U(x, µ) = W(ω) → U0(x, µ) = Λ(x) U(x, µ) Λ(x + aµˆ)−1 = W(ω0)
W(ω) → W(ω0(ω)) ,
(4.41)
(4.42)
we want dU0(x, µ) = dU(x, µ) . (4.43)
Q
N2−1
A naive expectation would be that dU ∝ dωa, but it is not quite correct because SU(N) a=1 is a curved manifold. Eg. for SU(2) a parametrization is the following.
W = w0 + iwkτk , wµ ∈ R , wµwµ = 1 .
(4.44)
So the manifold is a 3-sphere. It has a curvature. We may choose (these ω are not the ω of exp(ωaTa)
√
W = 1 − ωaωa + iωaτa , ωa ∈ [0, 1] .
(4.45)
To account for the curvature we deﬁne a metric tensor on the manifold
∂W ∂W†
∂ωa ∂ωb
Gab = tr ( (4.46)
) = Gba , G ≥ 0 , ← exercise
In a transformation ω → ω0 it changes to
∂W ∂W†
∂ω0a ∂ω0b
∂W ∂W† ∂ωc ∂ωd
∂ωc ∂ωd
∂ω0a ∂ω0b
Gab(ω0) = Tr ( ) = Tr ( ).(4.47)
= Gcd(ω)
∂ωc ∂ωd ∂ω0a ∂ω0b
As a remark (we do not need it): an invariant line element is
∂ωa ∂ωb
∂ω0c ∂ω0d ds2 = Gabdωadωb = Gab(ω) (4.48) dω0c dω0d = Gcd(ω0)dω0cdω0d .
Now take
N2−1
YpdW(ω) = det(G) (4.49) dωa . a=1
The pieces transform as
∂ωa
∂ω0b eq. (4.47) det(G0) det(G)(det( ))2 (4.50)
=
N2−1 N2−1
∂ω0a
∂ωb
YYdω0a =det( )dωc (4.51)
a=1 c=1
→dW(ω0) = dW(ω) . (4.52)
30 So we choose
N2−1
Y
√dU(x, µ) = C (4.53) dωa det G
a=1
ZdU = 1 (4.54) ﬁxes C .
This is the Haar measure, the invariant measure on the group.
Exercise
Show that for our SU(2)-parametrization
ωaωb + (1 − ωcωc) δab
Gab =,(4.55)
(1 − ωcωc)
Y
1
π2 const. × (1 − ωcωc)−1/2 dU(x, µ) =(4.56) dωa =
δ(1 − wµwµ)d4w . a
So the parameter space of the group is a 3-sphere.
4.2.1 Some group integrals (for later)
What about
ZZZhUi ≡ dU U = d(Λ−1U) U = dU0 ΛU0 = ΛhUi
(4.57)
(4.58) for all Λ ∈ SU(N) . In particular for
Λ = exp(i 2πn/N) ≡ exp(i 2πn/N) , n = 0 . . . N − 1 , the center of SU(N) then
N−1
X
1hUi = exp(i 2πn/N)hUi = hU†i = 0 . exp(i 2πn/N)hUi = 0
(4.59)
(4.60)
Nn=0
A non-trivial group integral:
ZZfijkl =dU Uij(U†)kl = dU Uij(Ulk)∗ (4.61)
ZZd(Λ−1U) Uij(U†)kl =(4.62) =
d(UΛ ) Uij(U )kl
−1
†
˜
ZZdU (ΛU)ij(U†Λ†)kl =(4.63) =
dU (UΛ)ij(Λ U )kl.
−1 †
˜˜
For:
ﬁxed j, k: Fil = fijkl ,(4.64)
ΛFΛ−1 = F ⇒ F = c
|{z }exercise
˜ ˜−1
ΛGΛ = G ⇒ G = c0
ﬁxed i, l: Gjk = fijkl ,(4.65)
⇒fijkl = c δilδjk (4.66)
31 It remains to determine c:
Z
XXfijjl =dU δil = c (4.67)
δilδjj
(4.68) jj
→ c = 1/N .
Exercise
F is an N × N matrix. Prove that if ΛFΛ−1 = F holds for all Λ ∈ SU(N) then F = c .
Hints: Start with N = 2. Find two special SU(2) matrices which allow to show F = c .
Embed SU(2) in SU(N) and use the N = 2 property to show it for all N.
4.3 Pure gauge theory
The path integral is now
Z
Y
G[U]
Z = D[U] e−S ,D[U] = dU(x, µ) , (4.69)
x,µ
Z
1
G[U] hO[U]i =
D[U] O[U] e−S (4.70)
Z
U(x + Lννˆ, µ) = U(x, µ) , PBC (4.71)
Λ(x + Lννˆ) = Λ(x) , for the gauge transformations (4.72) where S[U] = SG[U] when matter ﬁelds are neglected. This is the pure gauge theory. We will always ﬁrst work with the ﬁnite volume theory and then discuss the limit of large volume.
4.3.1 Gauge invariance
Consider some observable, O[U], any polynomial of the ﬁelds U(x, µ). Its expectation value is the same as the expectation value of any gauge transform of it:
Z
Z
Z
Y
1hO[U]i = dU(x, µ) O[U] e−S[U] (4.73)
(4.75)
Zx,µ
Y
1
−1
=
=(4.74) dUΛ (x, µ) O[U] e−S[U]
Zx,µ
Y
1dU(x, µ) O[UΛ] e−S[U] = hO[UΛ]i .
Zx,µ
We may deﬁne a projector onto the gauge invariant part of O[U]:
Z
Y
P0O[U] = dΛ(x) O[UΛ] . (4.76)
(4.77) x
Then integrating above over Λ(x) we see that hP0O[U]i = hO[U]i ,
32 so one needs to consider only the gauge invariant part of any observable. Consider in particular an observable (e.g. a parallel transporter from y to x) which transforms as