Notes on Quantum Field Theory

Andrew Forrester January 28, 2009

Contents

1 Questions that should be answered in an intro to QFT 3

2 Questions and Ideas 3

3 The Big Picture 3

4 Notation 4

5 Mathematics 4

6 Terms 5

7 Postulates and Theorems 5

8 New “Developing Quantum Field Theory” 58.1 Necessity for Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.2 Three Pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.3 From Lagrangians to Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

9 Initial “Developing Quantum Field Theory” 79.1 Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79.2 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99.3 Operators and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

9.3.1 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109.3.2 Bar Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119.3.3 The Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119.3.4 The Helicity Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

9.4 Equations of motion: A Unified View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129.5 The Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129.6 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9.6.1 The Quantized Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.6.2 Dirac Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.6.3 How Not to Quantize the Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . 159.6.4 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

9.7 Path-Integral Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.7.1 Feynman Diagrams and Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

10 Renormalization and “Regularization” 1610.1 Divergences, Power Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1710.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

11 Remarks 2011.1 Notable Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2011.2 Conflicting Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1

12 An Interpretive Introduction to Quantum Field Theory 2112.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.2 Preliminaries and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.3 From Particles to Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2

1 Questions that should be answered in an intro to QFT

• What’s the best motivation for QFT? (And other questions in the RFC.)• If QFT is an approximation to free field theory, how good is QFT at describing non-free systems such

as an atom or the field of a stationary particle?

2 Questions and Ideas

• “The spin statistics theorem, provable in QFT, asserts that particles with (magnitude of spin) equal toan even multiple of ~/2 are bosons, and those with spin equal to an odd multiple of ~/2 are fermions.”(Does not apply in 1D.) ([12] pg 273)Prove it!“The origin of this [theorem]. . . lies in the mathematical properties of the Lorentz transformations.”([13] pg 158, See section 13.4)“It seems bizarre that relativity should have anything to do with it, and there has been a lot of discussionrecently as to whether it might be possible to prove the spin-statistics connection in other (simpler)ways. See. . . ” ([10] pg 204)• Peskin’s explanation of Noether current doesn’t make sense. Where does jµ come from? How do we

explain its physical significance?• Goldstein’s explanation of Noether’s theorem is too involved. Simplify it!• Typo? (Peskin p. 61: “From this viewpoint, a positron is the absence of a negative-energy electron.”)

Shouldn’t that say “positive-energy electron”?• How does λ

n!φn correspond to an n-leg vertex interaction? How does −igψγ5ψφ correspond to a

. . . interaction?

Books

• Best introduction: [1] Hatfield• Good: [2] Ryder• Useful: [4] Zee• A few good ideas: [8] Teller• Peskin and Schroeder [3] covers

− weakly interacting particles and fields, using series expansions in the strength of the interaction

but does not cover

− history and experimental evidence of QFT− the theory of bound states− phenomena associated with nontrivial solutions to nonlinear field equations

3 The Big Picture

Physics 230A-C: “Modern quantum field theory, including quantum electrodynamics and quantum chromo-dynamics, renormalization group methods, path-integral quantization, spontaneous symmetry breakdown,monopoles and other solitons.”

3

Quantum Field Theory (QFT), Relativistic Quantum Mechanics What is it? Furthering quantummechanics somehow to create the Standard Model...

Subfields

• QFFT: quantum free-field theory(Unphysical pre-model for reality, with interactions “turned off”; develops physical ideas and formalism)• QED: quantum electrodynamics• QFD: quantum flavordynamics• QCD: quantum chromodynamics

Superfields(?)

• Standard Model of particle physicsWhat does the Standard Model include that may not be a part of QFT?

4 Notation

We use “natural” units, where ~ = c = 1, for most expressions.We use the “timelike convention” of metric signature (1,−1,−1,−1), meaning that the flat, Minkowski

spacetime metric is η = diag(1,−1,−1,−1), that is,

ηµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

µν

.

Since gravitation will not be incorporated, the general metric equals the Minkowski metric: gµν = ηµν .(Perhaps we could also say that all processes we examine will occur on small spatial scales, where we canalways boost to a reference frame where this equation is true, and gravitation will be irrelevant.)

xµ = (x0,x)µ xµ = ηµνxν = (x0,−x)µ

p ·x = ηµνpµxν = p0x0 − p ·x

A particle of mass m hasp2 = p · p = E2 − |p|2 = m2

Also 1

∂2 = ∂µ∂µ = ∂0

2 − ∂12 − ∂2

2 − ∂32 = 1

c2∂t

2 − ∂x2 − ∂y2 − ∂z2 = ∂02 −∇2.

5 Mathematics

QFT Math - functional analysis, theory of distributions (generalized functions), C∗-algebras, von Neu-mann algebras of bounded operators

Functional Derivatives and the Generating Functional

δf(y)δf(x)

= δ(x− y) orδ

δf(x)

∫dy f(y)φ(y) = φ(x)

1Sometimes ∂2 is written as �, in analogy to ∆, and sometimes it’s written as �2, in analogy to ∇2, usually in physicssettings. I prefer the �2 notation because it reveals its squared nature (while the box itself reveals its four-variable nature).

4

6 Terms

• Field Theory - local field theory ([4] Zee) (Lagrangian density versus Lagrangian)Free fields, Generalized free fields (mentioned in [7]), (bare mass, mentioned by Teller [8]), Interactingfields• Quantum Mechanics - Heisenberg’s QM, canonical QM, path-integral QM, axiomatic QM? axiomatic

QFT (S-Matrix versus local QFT)From [7]: “The original problem of axiomatic field theory, which has not yet been fully solved, wasto pick out and to formulate unambiguously the more trustworthy features of the formal apparatusassociated with the Lagrangian or Hamiltonian formalism. Therefore, for a critical reading of this bookand for an understanding of the inductive origin of the fundamental postulates of the theory one musthave some conception of the material consituting the ‘classical’ quantum theory of fields.”• Wave Mechanics -• Quantum Field Theory -• Time Ordering -• Minkowski Quanta -• Rindler Quanta -• Microcausality -• -• Feynman Diagram or Graph -• n-Point Function - (a blob with n legs)? (you may specify what particle is associated with each leg,

correct? as in photon n-point function [Peskin pg 318], or an electron 5- and photon 3-point function[?])A Green function?• -• π - Momentum Density Conjugate to ϕ

7 Postulates and Theorems

• Haag’s Theorem - states the interaction picture cannot be rigorously defined in a relativistic quantumfield theory. The theorem is inconvenient, however, since the canonical development of perturbativequantum field theory, which includes quantum electrodynamics – one of the great successes of modernscience, relies entirely on the interaction picture.• Furry’s Theorem - (Peskin pg 318) The photon three-point function vanishes.(?)

8 New “Developing Quantum Field Theory”

8.1 Necessity for Quantization

8.2 Three Pictures

8.3 From Lagrangians to Feynman Rules

(from Griffiths [9] particles book, page 358)Free Lagrangian ⇒ propagatorInteraction terms ⇒ vertex factors

5

Equations of motion: (should just write the operators p and x)[i~ ∂t +

~2

2m∇2 − V

]ϕ = 0 (Schrodinger eqn, for any/neglecting spin?)[

i~ ∂t +~2

2m∂x

2

]ϕ = 2c |ϕ|2 ϕ (Nonlinear Schro. Model; nonrel. Bose gas)[

i~ ∂t −1

2m{σ · (−i~∇− qA)

}2 − qΦ]ψ = 0 (Pauli eqn, for spin-1/2 2-spinors)[

∂µ∂µ +(mc

~

)2]ϕ = 0 (Klein-Gordon eqn, for spin-0)[

∂µ∂µ +(mc

~

)2]ϕ = − λ

3!ϕ3 (“Phi Fourth” eqn)[

iγµ∂µ −(mc

~

)]ψ = 0 (Dirac eqn, for spin-1/2 4-spinors)[

∂µ (∂µAν − ∂νAµ) +(mc

~

)2

Aν

]= 0 (Proca eqn, for spin-1)

[i~ ∂t −

(−i~∇− qA)2

2m− qΦ +

q~2mσ ·B

]ψ = 0 (“full Pauli eqn”, with Stern-Gerlach term)?

• Spin-0 propagator: ip2−(mc)2

• Spin-12 propagator: i

�p−mc= i(�p+mc)

p2−(mc)2

• Spin-1 propagator: −ip2−(mc)2

[gµν − pµpν

(mc)2

]Look at “List of quantum field theories” From Wikipedia, the free encyclopedia

• Chern-Simons model• Chiral model• Gross-Neveu• Kondo model• Lower dimensional quantum field theory• Minimal model• Nambu-Jona-Lasinio• Noncommutative quantum field theory• Nonlinear sigma model• Phi to the fourth• Quantum chromodynamics• Quantum electrodynamics• Quantum Yang-Mills theory• Schwinger model• Sine-Gordon• Standard model

6

• String Theory• Thirring model• Toda field theory• Topological quantum field theory• Wess-Zumino model• Wess-Zumino-Witten model• Yang-Mills• Yang-Mills-Higgs model• Yukawa model

From Mathworld:Quasilinear Klein-Gordon equation (Nayfeh 1972, p. 76; Zwillinger 1997, p. 133):[

∂t2 − α2∂x

2 + γ2]ϕ = βϕ3

Nonlinear Klein-Gordon equation (Matsumo 1987; Zwillinger 1997, p. 133):

n∑i=1

∂xi2ϕ = −λϕp

9 Initial “Developing Quantum Field Theory”

Sean Carroll[11] pg 41: “You may wonder what the purpose of introducing a Lagrangian formulation is, ifwe were able to invent the equations of motion before we ever knew the Lagrangian (as Maxwell did forhis equations). There are a number of reasons, starting with the basic simplicity of positing a single scalarfunction of spacetime, the Lagrange density, rather than a number of (perhaps tensor-valued) equations ofmotion. Another reason is the ease with which symmetries are implemented; demanding that the actionbe invariant under a symmetry ensures that the dynamics respects the symmetry as well. Finally, as wewill see in Chapter 4, the action leads via a direct procedure (involving varying with respect to the metricitself) to a unique energy-momentum tensor.”

Advantages of Lagrangian/Hamiltonian formalism over mere positing of equations of motion:• Simplicity of positing a single scalar function of spacetime• The ease with which symmetries “are implemented”: symmetry transfers to equations of motion• Action leads via direct procedure to a unique stress tensor• (Possibly leads to other conserved quatities too, right?)

9.1 Lagrangians

List:• Lagrangian (density) and physical interpretation• Equation of Motion• Symmetries and Conserved Quantities

L = −mc2√

1− β2 − qΦ +q

cv ·A +

∫dV L0

7

Basic Field Lagrangians

Schrodinger (which is free)

LSchro =i

2(ϕ∗∂tϕ− ϕ∂tϕ∗

)− 1

2(∂xϕ∗)(∂xϕ) + V (x)ϕ∗ϕ

HSchro =12

(∂xϕ∗)(∂xϕ) + V (x)ϕ∗ϕ

Schrodinger with 2-body interaction

H2-intSchro =

12

(∂xϕ∗)(∂xϕ) + V (x)ϕ∗ϕ+ V (x)ϕ∗ϕ∗ϕϕ

Schrodinger with 3-body interaction

H3-intSchro =

12

(∂xϕ∗)(∂xϕ) + V (x)ϕ∗ϕ+ V (x)ϕ∗ϕ∗ϕ∗ϕϕϕ

Nonlinear Schrodinger ModelHNLS = ϕ∗(−∂x2)ϕ+ cϕ∗ϕ∗ϕϕ

Klein-Gordon (which is free)

LK-G =12

(∂µϕ)(∂µϕ)− 12m2ϕ2

HK-G =12π2 +

12|∇ϕ|2 +

12m2ϕ2

Free Dirac

LfreeDirac = ψ(iγµ∂µ −m)ψ

= ψ(i�∂ −m)ψ

QED

Full Lagrangian (Peskin pg 303):

LQED = ψ(i��D−m)ψ − 14(Fµν)2

= ψ(i�∂ − e��A−m)ψ − 14(Fµν)2

= ψ(iγµ∂µ −m)ψ − eψγµψAµ − 14(Fµν)2

= LfreeDirac − eψ��Aψ − 1

4(Fµν)2

Complex scalar field interacting with the electromagnetic field Aµ (Peskin pg 312):

L = −14(Fµν)2 + |Dφ|2 −m2 |φ|2

Phi-Fourth Theory

Lφ4 =12

(∂µφ)2 − 12m2φ2 − λ

4!φ4

Yukawa Theory

Pseudoscalar Yukawa Lagrangian (Peskin pg 344):

L = 12(∂φ)2 − 1

2m2φ2 + ψ(i�∂ −M)ψ − igψγ5ψφ

8

EW Theory

L1 gen, no HiggsEW =

(νe e

)Li��D

(νe

e

)L

+(u d

)Li��D

(ud

)L

+eRi��DeR + uRi��DuR + dRi��DdR

(+νeRi��DνeR

)− 1

4Wµν

iWµνi −14Y µνYµν

where e, νe, u, and d are the fields for the (left- and right-handed) quarks, Wi and Y are the potentials forthe gauge bosons, and

Wµνi ≡ ∂µW ν

i − ∂νW νi + gεijkW

µjW

νk

Y µν ≡ ∂µY ν − ∂νY ν

Dµ = ∂µ − igWτi2Wµi − ig′Y Yµ

where Y is the hypercharge (or “isocharge”?) and W is the “chiral charge” (or “ultracharge”)

Others

• Schrodinger-Newton equations (Roger Penrose’s gravitationally-induced wave-function collapse scheme)

What’s This?

• Transformations of the field variables only

− Gauge transformations of the first kind (δxµ = 0, δηρ = εcρηρ, no summation, where the cρ areconstants)

− Gauge transformations of the second kind (such as adding the four-gradient ∂µΛ to Aµ)

• “weak” constraints (see Goldstein, pp. 329, 583 (footnote))From Abers, page 47, we have that the classical equation of motion is

A(t) = {A,H}+ ∂t

9.2 Quantum Field Theory

Two Formalisms:• Path-integral formalism ([4] Zee pg 61: “the quickest way to QFT”)• Canonical formalism• See [4] Zee pp 61 (“Complementary formalisms”) and 67 (“Nobody is perfect”)

One idea:• Posit Lagrangian• Derive Equation of motion for free field• Derive conserved quantities for free field and use these to come up with interaction equations... of

motionOther thoughts:• Posit equation of motion (of what? Wavefunction or Field? Wavefunction/Field? for free field only?)• Derive conserved quantities (for free field?) (Use these quantities to come up with interaction equa-

tions... of motion?)

9

• Show certain properties (e.g., that creation operators create states with half-integer spin) using thefact that momentum is the conserved quantity for translations, angular momentum is the conservedquantity for rotations, energy is the conserved quantity for time translations

9.3 Operators and Transformations

Active scalar transformationφ(x)→ φ′(x) = φ(Λ−1x)

Active n-component multiplet linear transformation

Φa(x)→Mab(Λ) Φb(Λ−1x)

Φ→M(Λ) Φ

Φ→M(Λ′)M(Λ) Φ = M(Λ′′) Φ

for Λ′′ = Λ′ΛWhere does this come from?:

[Jµν , Jρσ] = i(gνρJµσ − gµρJνσ − gνσJµρ + gµσJνρ)

A particular representation:(J µν)αβ = i(δµαδνβ − δµβδνα)

9.3.1 Gamma Matrices

Four n× n matrices γµ satisfying the anticommutation relations

[γµ, γν ]+ ≡ γµγν + γνγµ = 2gµν × 1n

n-dimensional representation of the Lorentz algebra:

Sµν =i

4[γµ, γν ]

In 3-D Euclidean space we have

γj ≡ iσj (Pauli sigma matrices)

[γi, γj ]+ = −2δij

The i by the Pauli sigma matrix and the negative sign by the Kronecker delta are purely conventional. So

Sij =12εijkσk

In 4-D Minkowski space we have (in the Weyl or chiral representation, and block-matrix form)

γ0 =

(02 12

12 02

)abuse=

(0 11 0

)

γi =

(02 σi

−σi 02

)

10

S0i =i

4[γ0, γi] = − i

2

(σi 00 −σi

)

Sij =i

4[γi, γj ] =

i

2εijk

(σk 00 σk

)≡ 1

2εijkΣk

A four-component field ψ that transforms under boosts and rotations according to the above two equationsis called a Dirac spinor. (“ψ is not a wavefunction; it is a classical field,” pg 41 ...for now... I think)

Also[γµ, Sρσ] = (J ρσ)µνγ

ν

or, equivalently,(1 + i

2ωρσSρσ)γµ(1− i

2ωρσSρσ) = (1− i

2ωρσJρσ)µ

νγν

which is the infinitesimal form ofΛ−1

12

γµΛ 12

= Λµνγν

where

Λ 12

= exp(− i

2ωµνS

µν

)(the spinor representation of the Lorentz transformation Λ)

9.3.2 Bar Notation

ψ ≡ ψ†γ0 = ψ†

(02 12

12 02

)

σµ ≡ (12,σ) and σµ ≡ (12,−σ)

γµ =

(02 σµ

σµ 02

)Later

u(p) = u†(p)

9.3.3 The Lorentz Transformation

The general representation:

D(Λ) = exp(− i

2ωαβM

αβ

)where ωαβ = −ωβα are the transformation parameters, and Mαβ are the six generators of the (properorthochronous) Lorentz group.

9.3.4 The Helicity Operator

h ≡ p ·S =12pi

(σi 00 σi

)

11

9.4 Equations of motion: A Unified View

[4] Zee pg 91: “. . . a unified view of the equations of motion in relativistic physics: They just project outthe unphysical components. . . . The Klein-Gordon equation (∂2 + m2)ϕ(x) = 0 just projects out thoseFourier components ϕ(k) not satisfying the mass shell condition k2 = m2.” “. . . the equation of motion ofa massive spin-1 particle . . . one of the four components of Aµ is projected out.”

[4] Zee pg 91: “. . . the mysterious Dirac equation is no more and no less than a projection that gets ridof the unwanted degrees of freedom.” (4◦ to 2◦)

[4] Zee pg 115: “. . . deep group theoretic way of looking at the Dirac equation: It is a projection boostedinto an arbitrary frame.”

9.5 The Klein-Gordon Equation

(E2 −

[p2c2 +m2c4

])φ = −~2c2

(∂2 + ν2

)φ = 0

where E and p indicate the relativistic energy and momentum operators, ∂2 = � = 1c2∂t

2 − ∇2, andν ≡ mc/~ (Compton inverse wavelength or spatial frequency).

[ap, a†p] = (2π)3 δ(p− p′)

H =∫ d3p

(2π)3ωp

(a†p ap + 1

2 [ap, a†p])

=∫ d3p

(2π)3ωp a

†p ap

neglecting the common term that would cause the integral to diverge (as it yeilds δ(0))

φ(x) =∫ d3p

(2π)31√2ωp

(ap e

ip ·x + a†p e−ip ·x

)=∫ d3p

(2π)31√2ωp

(ap + a†−p

)eip ·x

π(x) = (−i)∫ d3p

(2π)3

√ωp

2

(ap e

ip ·x − a†p e−ip ·x)

= (−i)∫ d3p

(2π)3

√ωp

2

(ap − a†−p

)eip ·x

[φ(x), π(x′)] = i~ δ(x− x′)

[φ(x), φ(x′)] = [π(x), π(x′)] = 0

9.6 The Dirac Equation

[4] Zee pg 91: “. . . the mysterious Dirac equation is no more and no less than a projection that gets rid ofthe unwanted degrees of freedom.” (4◦ to 2◦)

[4] Zee pg 115: “. . . deep group theoretic way of looking at the Dirac equation: It is a projection boostedinto an arbitrary frame.”Dirac Equation (of motion):

(γµpµ −m)ψ ≡ (�p−m)ψ = 0

12

4D field Dirac equation:

∂t

ψ3

ψ4

ψ1

ψ2

− ∂x

ψ4

ψ3

−ψ2

−ψ1

− i∂y−ψ4

ψ3

ψ2

−ψ1

− ∂z

ψ3

−ψ4

−ψ1

ψ2

± im

ψ1

ψ2

ψ3

ψ4

=

0000

Planewave solutions:Positive-frequency (“monochromatic”) solutions:

ψ(x) = u(p) e−ip ·x

Negative-frequency (“monochromatic”) solutions:

ψ(x) = v(p) e+ip ·x

where u(p), v(p) (∈ C2?) are column vectors and p2 ≡ pµpµ = Ep2 − p2 = m2.

u(p0) =√m

(ξξ

)=

√mc

(ξξ

) ?

where p0 ≡ (m,0) and ξ (∈ C2?) is a two-component spinor (for example, ξ =

(10

)is the spin-up spinor,

with spin along the 3-direction). ∑s=1,2

ξsξs† = 12 =

(1 00 1

)

us(p) =

( √p ·σ ξs

−√p · σ ξs

)

vs(p) =

( √p ·σ ηs

−√p · σ ηs

)s ∈ {1, 2}, where ξs and ηs are bases of two-component spinors.

(“Spin sum”?) orthogonality relations (INNER PRODUCTS): (I think “spin sum” implies outer prod-uct.)

ur(p)us(p) = 2mδrs or ur†(p)us(p) = 2Ep δrs

vr(p) vs(p) = −2mδrs or vr†(p) vs(p) = 2Ep δrs

Spin sum completeness relations (quoting from pg 53, eqns 3.66, 3.67) (OUTER PRODUCTS):

∑s=1,2

us(p) us(p) = γ · p+m = �p+m =

(m p ·σp · σ m

)

∑s=1,2

vs(p) vs(p) = γ · p−m = �p−m =

(−m p ·σp · σ −m

)Be careful:

ur†(p) vs(p) 6= 0 and vr†(p)us(p) 6= 0

(not orthogonal) and note:ur†(p) vs(p) = vr†(p)us(p) = 0

13

9.6.1 The Quantized Dirac Field

H =∫

d3p

(2π)3∑s

Ep

(asp† asp + bsp

† bsp

)[arp, a

sq†]+ = [brp, b

sq†]+ = (2π)3 δ(p− q) δrs

All other anticommutators equal zero:

[arp, asq]+ = [arp

†, asq†]+ = [brp, b

sq]+ = [brp

†, bsq†]+ =

[arp, asq†]+ = [arp, b

sq]+ = [arp, b

sq†]+ =

[arp†, bsq]+ = [arp

†, bsq†]+ =

[brp, bsq†]+ = 0

Commutators?[arp† arp, a

s0†] = arp

† arp as0† − as0

† arp† arp = (2π)3 δ(p) ar0

† δrs

The vacuum state |0〉 is defined to be the state such that

asp |0〉 = bsp |0〉 = 0

P =∫

d3xψ†(−i∇)ψ =∫

d3p

(2π)3∑s

p(asp† asp + bsp

† bsp

)The one-particle states

|p, s〉 ≡√

2Ep asp† |0〉

are defined so their inner product

〈p, r|q, s〉 = 2Ep(2π)3 δ(p− q) δrs

is Lorentz invariant.

ψ(x) =∫ d3p

(2π)31√2Ep

∑s=1,2

(asp u

s(p) e−ip ·x + bsp† vs(p) eip ·x

)=∫ d3p

(2π)31√2Ep

∑s=1,2

(asp u

s(p) + bs−p† vs(−p)

)e−ip ·x

ψ(x) =∫ d3p

(2π)31√2Ep

∑s=1,2

(bsp† vs(p) e−ip ·x + asp u

s(p) eip ·x)

=∫ d3p

(2π)31√2Ep

∑s=1,2

(bsp† vs(p) + as−p u

s(−p))e−ip ·x

At t = 0, we have

ψ(x) =∫ d3p

(2π)31√2Ep

∑s=1,2

(asp u

s(p) eip ·x + bsp† vs(p) e−ip ·x

)=∫ d3p

(2π)31√2Ep

∑s=1,2

(asp u

s(p) + bs−p† vs(−p)

)eip ·x

14

ψ(x) =∫ d3p

(2π)31√2Ep

∑s=1,2

(bsp† vs(p) eip ·x + asp u

s(p) e−ip ·x)

=∫ d3p

(2π)31√2Ep

∑s=1,2

(bsp† vs(p) + as−p u

s(−p))eip ·x

[ψ(x), ψ†(y)]+ = δ(x− y) 14

[ψa(x), ψ†b(y)]+ = δ(x− y) δab

From Noether’s Theorem, we can derive

J =∫

d3xψ†(

x× (−i∇) +12Σ)ψ

In the rest frame,

Jz =∫

d3xψ†12

Σ3ψ

9.6.2 Dirac Propagator

9.6.3 How Not to Quantize the Dirac Field

ψ(x) =∫

d3p

(2π)31√2Ep

eip ·x∑s=1,2

(asp u

s(p) + bs−p vs(p)

)[arp, a

sq†] = [brp, b

sq†] = (2π)3 δ(p− q) δrs

9.6.4 Electromagnetic Field

[4] Zee pg 177: “According to the gauge principle already used to write the Schrodinger’s equation in anelectromagetic field, to obtain the Dirac equation for an electron in an external electromagnetic field wemerely have to replace the ordinary derivative ∂µ by the covariant derivative Dµ = ∂µ − ieAµ:”(

iγµDµ −m)ψ = 0

9.7 Path-Integral Formalism

D + 1 spacetime dimensions:

Z(J) =∫

Dϕ eiR

dD+1xn

12 [(∂ϕ)2−m2ϕ2]−λϕ4+Jϕ

o

evaluate the integral as a series in λ:

∞∑k=0

(−iλ)k

k!

∫Dϕ ϕ(x1)ϕ(x2) · · ·ϕ(xn)

[∫dD+1y ϕ(y)4

]keiR

dD+1xn

12 [(∂ϕ)2−m2ϕ2]−λϕ4

o

(See [4] Zee pg 84: Field Theory Redux)Free Field Theory:Interaction:

Z(J) =∫

Dϕ eiR

d4xn

12 [(∂ϕ)2−m2ϕ2]− λ

4!ϕ4+Jϕ

o

15

(From [4] Zee pg 41) For simplicity, we add only one anharmonic term (− λ4!ϕ

4) to our free field theoryto destroy the linearity of the equations and allow for interaction and scattering of vibrational modes.(Footnote: “Thanks to the propagation of ϕ, the sources coupled to ϕ interact, . . . but the particlesassociated with ϕ do not interact with each other. This is like saying that charged particles coupled to thephoton interact, but (to leading approximation) photons do not interact with each other.)Quantum Mechanics (D = 0):

Z(J) =∫

Dϕ eiR

dt

(12

»(dϕdt )2−m2ϕ2

–−λϕ4+Jϕ

)

9.7.1 Feynman Diagrams and Rules

(Explain the difference between a diagram (or a specific probability amplitude) and the diagram series inperturbative expansion.)• Feynman propagators for each internal line (or each line?)• What do the dots mean? Can you have an “amputated” leg without a dot on one end?• Feynman Diagram or Graph -

− Line - internal, external (internal/external momentum)− Vertex -− (External) Point -− Loop -− Leg -− Amputated Leg -− Propagator - photon propagator, electron propagator− Arrow - Direction of momentum transfer, Direction of current (charge flow)− Blob -− Subdiagram - reducible diagram, one-particle-irreducible diagram (Peskin pg 317)− -

10 Renormalization and “Regularization”

• Why “re” normalization?From http://math.ucr.edu/home/baez/lengths.html#bohr_radius“Indeed, renormalization was an issue in classical field theory before quantum field theory came along.”“Renormalization is an aspect of field theory which deals with such issues as the fact that the electro-

magnetic field produced by an electron has energy and thus should be counted as part of the mass of theelectron!”

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10.1 Divergences, Power Counting

Superficial divergence D ≡ (power of momentum in numerator)− (power of momentum in denominator)

d = spacetime dimension for the field theoryn = power of (an) interaction for the field theoryL = number of lines (which can be solid, dashed, curly, etc.)

X,N = number of external lines (“legs”? and “points”? as in N -Point Function?)I = number of internal lines (and propagators)V = number of vertices` = number of loopsL = I − V + 1 (topological relation)

(number of points = number of unamputated legs?)• Purely scalar field theory with φn interaction term: L = 1

2(∂φ)2 − 12µ

2φ2 − λn!φ

n

− Any dimension d: D = d+[n(d−22

)− d]V −

(d−22

)N

• QED: D = d`− Ie − 2Iγ

− 4D: D = 4−Xγ − 32Xe (dependent only on external legs)

− Any dimension d: D = d+(d−42

)V −

(d−12

)Xe −

(d−22

)Xγ

Ultraviolet Behavior(?) (Peskin pg 321) (“counting of ultraviolet divergences”)• (Superficially) Super-Renormalizable theory• (Superficially) Renormalizable theory• (Superficially) Non-Renormalizable theorydc = critical dimension where the theory can be renormalized (above this, non-renormalizable)What about Infrared Behavior?

10.2 Methods

• Momentum Cutoff: integrate from zero to Λ rather than infinity

• Dimensional Regularization: d→ d+ ε, where ε→ 0That’s dimension as in 4D, not dimension as in quantity dimensions.

− with minimal subtraction. . .

• Minimal Subtraction – Terms and Counter-Terms (CT) (pg 15 in notes)

− Simply cancel out the pole (a divergent “term” or graph) by adding a CT that has the samedegree of divergence (in a subgraph, at least, right?)

• Equivalent methods with different bookkeeping (Peskin pg 326):

− Bare perturbation theory− Renormalized perturbation theory

• BPH scheme (Bogoliubov-Parasink-Hepp ?), R-procedure

− Taylor expansion of. . . ;( iterative procedure, right?)

17

− renormalized amplitude: R(AG) of a graph G

− R(AG) ≡ {AG with all subdivergences subtracted}− renormalized part: a 1PI (one-particle irreducible) graph w/ D(G) ≥ 0

• “. . . an alternative regularization/renormalization method that is applicable in condensed matterphysics” (pg 30 of class notes)

• Renormalization Group (RG)

− running coupling constants: m(µ), λ(µ)

∗ Γ(α)0

({pi}, λ0,m0,Λ or ε

)= Z(µ)−N/2 Γ(α)

({pi}, λ(µ),m(µ), µ

)(why α?) N = number of legs

⇒ Γ(N)({pi}, λ(µ′),m(µ′), µ′

)=(Z(µ)Z(µ′)

)−N/2Γ(N)

({pi}, λ(µ),m(µ), µ

)or Γ(N)

({pi}, λ(µ′),m(µ′), µ′

)=[Z(µ′, µ)

]N Γ(N)({pi}, λ(µ),m(µ), µ

)where Z(µ′, µ) ≡

(Z(µ′)/Z(µ)

)1/2− RG Eqn

∗ ddµΓ(N)

0 = 0 (Γ0 is independent of µ)

⇒[µ d

dµ −N2 γφ

]Γ(N)

({pi}, λ,m, µ

)= 0

where γφ(µ) ≡ µ ddµ lnZ(µ) = µ d

dµ ln(1 + δZ(µ)

)= −2µ d

dµ lnZ(µ, µ′)

⇒[µ ∂∂µ + β ∂

∂λ −m2γm

∂∂m2 − N

2 γφ

]Γ(N)

({pi}, λ,m, µ

)= 0

where β(µ) ≡ µ ddµλ(µ) γm(µ) ≡ −m−2µ d

dµm2(µ)

− Wilsonian defn of RG transformation (non-perturbative or exact RG):

(1) Course-graining (thinning of degrees of freedom, or integrating over momentum shell Λ→ κΛ)(2) Rescaling of unit of length(3) Rescale (“renormalize”) your field variable(s) (ensuring that kinetic energy term (∂ϕ)2 has

unit renormalization)

− (Sean: RG → semigroup; trajectory: you can take it one way but not the other)

More Stuff

• Renormalization Parameters and Constants(versus bare parameters(?): m and m0; q and q0, or λ and λ0; or, more generally, gi and gi0)

− They are fixed by renormalization conditions (e.g. Γ(2)(p2 = 0) = m2, Γ(4)(pi = 0) = λ, Γ(4)(pi2 =µ2) = −iλ where you are free to choose µ and λ is an implicit function of µ)

− (•) charge renormalization (λ0, λ, δλ)− (×) mass renormalization (m0, m, δm2)− (⊗) field or wavefunction renormalization (2-loop order) (Zφ, δZφ) (pg 12 in notes)

• mass-independent renormalization schemes versus mass-dependent ones

− mass-independent: dimensional regularization with minimal subtraction

Examples (of what?) in• φ3 in d = 6 (many places in notes)

18

• φ3 in d = 4 (pg 8 230B notes)• φ4 in d = 4 (many places in notes)

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11 Remarks

11.1 Notable Remarks

• “Renormalization Group. . . the most important part of field theory.” – Prof. Tomboulis

11.2 Conflicting Remarks

“The name ‘quantum field theory’ leads us to expect a theory that must primarily be thought of as a fieldtheory. The single most important point I hope to make in this book is that this impression is wrong.When described correctly, the theory is just as much a ‘particle’ theory.” Teller [8]

Versus“Expositions of quantum mechanics will occasionally make the point that waves and particles are comple-mentary notions with different domains of validity, but don’t be misled; in quantum field theory it is thefields that are truly fundamental, while the particles are approximate notions useful in certain restrictedcircumstances.” Carroll [11] pg 386

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12 An Interpretive Introduction to Quantum Field Theory

12.1 Questions

• How does adding 1√ω(k)

make the Fourier transform |x〉 =∫

d3k(2π)3/2

e−ik ·x |k〉 make it manisfestly

covariant?

12.2 Preliminaries and Overview

“The name ‘quantum field theory’ leads us to expect a theory that must primarily be thought of as a fieldtheory. The single most important point I hope to make in this book is that this impression is wrong.When described correctly, the theory is just as much a ‘particle’ theory.”

Theories and Thier Interpretation

• “observational terms”• “theoretical terms”• “systems”• laws

Laws are not eternal truths to be used onl as premises in deductions. Instead they are like basic dresspatterns, to be tailored to suit the idiosyncrasies of the different customers. Properly tailored lawswork together to form a model.• models

never expected to correspond exactly to physical systems;• theories

− Theories, in turn, are collections of models that have been loosely grouped both according to phe-nomena to be modeled and according to common technical tools used in building the models. On thisdescription of theories, their boundaries are not sharp, and groupong can be somewhat arbitrary.

− This way of thinking applies nicely to various prescientific, or extra-sciendtific, ways of thinkingabout the world. For example, ... we tend to think of phhysical objects as composed of someinfinitely divisible material substance in which properties “inhere” and which provides or fixes theobject’ identities. This description outlines a type of model that agrees, to various degrees, withthe nature of physical objects in certain ways and not at all in others.

− Indeed, this view of theories, as a kind of model of scientific theorizing, applies very nicley to itself.“interpretation of theories”

• “interpretation of theories”We are elaborating in any way that clarifies, sharpens, or extends the similarity relation between amodel (or models loosely thought of as comprising a theory) and the things described by the theory.realism... distinction between chairs and volt meters, and electrons and electric charges?• “unobservables” and “observables”? realism question

− instrumentalism: interpretation limited to parts of theories that are supposed to correspond to“observable” things

− constructive empiricism: softens this stand a little: say what seems plausible and useful about the“unobservables” but be agnostic about any literal reading of such parts of the similarity relations.

− full-blown realism:...− rejection of realism: there is no reasonable distinction between the observable and unobservable

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Two General Issues of Interpretation for Quantum Theories

• superposition?• properties• dispositions: a disposition’s display property, activating condition

− disposition of an object: a state or property that involves, or perhaps is constituted by, the reliablemanifestation of some further property, the display property, when activating conditions are ineffect.

− display property may be a disposition: litmus paper has the disposition to turn red in an acid bath,but being red itself is a disposition to appear in various ways in various circumstances.

• propensity, probability

− activating conditions: “measurement”? the measurement problem. (from the possible to the actual)− superposition: propensities or hidden variables?

Overview of This Book

• quantum field theory: a particle and field (wave) theory• particle versus quantum

− Prequantum notions: A particle always has an exact space-time trajectory. Particles are thoughof as substantial - comprised of bits of substance or “stuff” in which properties can inhere. Inaddition, we often think of the infividual substance of a given particle as providing the particle withits ultimate identity, its being “this one” as opposed to “that one.”

− Quantum notions: Conventional quantum mechanics gives up on exact trajectories - the uncertaintyrelations for position and momentum require these never to receive simultaneous exact values inquantum descriptions of particles.

− Hilbert spaces: allow particle labels and counting (surplus formal structure, a burden of nonfunc-tioning theoretical machinery)

− Fock spaces: eliminates particle latbels and nonsymmetric states. A Fock space does not have thedesciptive machinery that would enable one to say which particle gets which property. It tells usonly what patterns of properties are exhibited. This facilitates a conception of what I will nowcall quanta, for which there is no “this one” as oppsed to “that one” independent of distinguishingproperties.

− Particles have primitive thisness, property-transcending individuality, but quanta do not. Thingswith primitive thisness can be counted; that is, we can think of the particles as being countedout, the first one, the second one, the third, and so on, with there being a difference in principlein the order in which they are counted, a difference that does not depend on which particle haswhich properties. By way of contrast quant can only be aggregated ; that is, we can only heapthem up in different quantities with a total measure of one, or two, or three, and so one, but inthese aggregations there is no difference in principle about which one has which properties. Thedifference between coutability and susceptibility to being merely aggregated is like the differencebetween pennies in a piggy bank and money in a modern bank account.

− What about “an indefinite number of quanta”? We usually think of properties as being inherentin things, and a prepensity, as a kind of disposition, is ordinarily thought of as a property. Whatthing has the propensity to display either one or two quanta when the activating conditions areset in motion? (It’s not the quanta. Perhaps call it “the world”. ...Why not wavefunction???Or wavefunction-system... ???) Or reject the assumption that instantiated properties are always

22

properties of things. (No!!!)

12.3 From Particles to Quanta

Ideas of Substance

•

The Labeled Tensor Product Hilbert Space Formalism (LTPHSF)

•

The LTPHSF and Primitive Thisness

•

Difficulty with Interpretation of the LTPHSF Using Primitive Thisness

•

Quanta

•• Spin-statistics theorem: describes connections among spin, relativity, and the relevant choice among the

commutators. It would be gratifying if an application of the formalism of the spin-statistics theorem tothe present interpretive framework enabled us to see in a more nearly physical and intuitive way hownature makes room for Fermions as well as Bosons.

23

References

[1] Brian Hatfield: Quantum Field Theory of Point Particles and Strings, Addison Wesley Longman, Inc.(1992)

[2] Lewis H. Ryder: Quantum Field Theory, Second Edition, Cambridge University Press (1996)

[3] Michael E. Peskin, Daniel V. Schroeder: An Introduction to Quantum Field Theory, Westview Press(1995)

[4] A. Zee: Quantum Field Theory in a Nutshell, Princeton University Press (2003)

[5] F. Mandl, G. Shaw: Quantum Field Theory, Revised Edition, John Wiley & Sons (1993)

[6] I. J. R. Aitchison, A. J. G. Hey: Gauge Theories in Particle Physics, A Practical Introduction, ThirdEdition. Volume I: From Relativistic Quantum Mechanics to QED, Taylor & Francis Group, LLC(2003)

[7] N. N. Bogoliubov, A. A. Logunov, I. T. Todorov: Introduction to Axiomatic Quantum Field TheoryW. A. Benjamin, Inc. (1975)

[8] Paul Teller: An Interpretive Introduction to Quantum Field Theory, Princeton University Press (1995)

[9] David J. Griffiths: Introduction to Elementary Particles, John Wiley & Sons (1987)

[10] David J. Griffiths: Introduction to Quantum Mechanics, Second Edition, Pearson Education, Inc.(2005)

[11] Sean M. Carroll: Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley(2004)

[12] Ramamurti Shankar: Principles of Quantum Mechanics, Plenum Press (1980)

[13] Ernest S. Abers: Quantum Mechanics, Pearson Education, Inc. (2004)

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