Advanced Quantum Field Theory Chapter 1 Canonical Quantization of Free Fields
Jorge C. Rom˜ao Instituto Superior T´ecnico, Departamento de F´ısica & CFTP A. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Fall 2013 Lecture 1
Canonical Quantization
Scalar fields
Lecture 2
Dirac field
Electromagnetic field
Discrete Symmetries Lecture 1
Jorge C. Rom˜ao TCA-2012 – 2 Canonical quantization for particles
❐ Let us start with a system that consists of one particle with just one degree Lecture 1 of freedom, like a particle moving in one space dimension. The classical Canonical Quantization • Quantization in QM equations of motion are obtained from the action, • Quantization in FT • Symmetries t2 Scalar fields S = dtL(q, q˙) . Lecture 2 Zt1 Dirac field Electromagnetic field ❐ The condition for the minimization of the action, δS = 0, gives the Discrete Symmetries Euler-Lagrange equations, d ∂L ∂L = 0 dt ∂q˙ − ∂q which are the equations of motion. ❐ Before proceeding to the quantization, it is convenient to change to the Hamiltonian formulation. We start by defining the conjugate momentum p, to the coordinate q, by ∂L p = ∂q˙
Jorge C. Rom˜ao TCA-2012 – 3 Quantization:The Hamiltonian and Poisson Brackets
❐ Then we introduce the Hamiltonian using the Legendre transform Lecture 1
Canonical Quantization • Quantization in QM H(p, q) = pq˙ L(q, q˙) • Quantization in FT − • Symmetries ❐ Scalar fields In terms of H the equations of motion are,
Lecture 2
Dirac field ∂H H, q PB = =q ˙ Electromagnetic field { } ∂p Discrete Symmetries ∂H H,p = =p ˙ { }PB − ∂q
❐ The Poisson Bracket (PB) is defined by
∂f ∂g ∂f ∂g f(p, q), g(p, q) = { }PB ∂p ∂q − ∂q ∂p obviously satisfying
p, q = 1 . { }PB
Jorge C. Rom˜ao TCA-2012 – 4 Quantization: The Schr¨odinger picture
❐ The quantization is done by promoting p and q to hermitian operators that Lecture 1 will satisfy the commutation relation (¯h = 1), Canonical Quantization • Quantization in QM • Quantization in FT [p, q] = i • Symmetries − Scalar fields ∂ Lecture 2 which is trivially satisfied in the coordinate representation where p = i . Dirac field − ∂q Electromagnetic field ❐ The dynamics is given by the Schr¨odinger equation Discrete Symmetries ∂ H(p, q) Ψ (t) = i Ψ (t) | S i ∂t | S i
❐ If we know the state of the system in t = 0, Ψ (0) , then the previous | S i equation completely determines the state Ψs(t) and therefore the value of any physical observable. | i ❐ This description, where the states are time dependent and the operators, on the contrary, do not depend on time, is known as the Schr¨odinger representation.
Jorge C. Rom˜ao TCA-2012 – 5 Quantization: The Heisenberg picture
❐ There exists and alternative description, where the time dependence goes to Lecture 1 the operators and the states are time independent. This is called the Canonical Quantization • Quantization in QM Heisenberg representation. • Quantization in FT • Symmetries ❐ To define this representation, we formally integrate to obtain Scalar fields Lecture 2 −iHt −iHt ΨS(t) = e ΨS(0) = e ΨH . Dirac field | i | i | i Electromagnetic field Discrete Symmetries ❐ The state in the Heisenberg representation, ΨH , is defined as the state in the Schr¨odinger representation for t = 0. The| unitaryi operator e−iHt allows us to go from one representation to the other. ❐ If we define the operators in the Heisenberg representation as,
iHt −iHt OH (t) = e OSe
then the matrix elements are representation independent. In fact,
Ψ (t) O Ψ (t) = Ψ (0) eiHtO e−iHt Ψ (0) h S | S| S i S | S | S = ΨH OH (t) ΨH . h | | i
Jorge C. Rom˜ao TCA-2012 – 6 Quantization: The Heisenberg picture
❐ The time evolution of the operator OH (t) is then given by the equation Lecture 1
Canonical Quantization • Quantization in QM dOH (t) ∂OH = i[H,OH (t)] + • Quantization in FT dt ∂t • Symmetries
Scalar fields The last term is only present if OS explicitly depends on time. Lecture 2 Dirac field ❐ In the non-relativistic theory the difference between the two representations Electromagnetic field is very small if we work with energy eigenfunctions. For the relativistic Discrete Symmetries theory, the Heisenberg representation is more convenient, because Lorentz covariance is more easily handled in the Heisenberg representation, because time and spatial coordinates are together in the field operators. ❐ In the Heisenberg representation the fundamental commutation relation is now
[p(t), q(t)] = i − ❐ The dynamics is now given by
dp(t) dq(t) = i[H,p(t)] ; = i[H, q(t)] dt dt Jorge C. Rom˜ao TCA-2012 – 7 Quantization: The Heisenberg picture
❐ Notice that in this representation the fundamental equations are similar to Lecture 1 the classical equations with the substitution, Canonical Quantization • Quantization in QM • Quantization in FT , P B = i[, ] • Symmetries { } ⇒ − Scalar fields Lecture 2 ❐ In the case of a system with n degrees of freedom the generalization is Dirac field Electromagnetic field [pi(t), qj(t)] = iδij Discrete Symmetries − [pi(t),pj(t)]=0
[qi(t), qj(t)]=0
and
p˙i(t) = i[H,pi(t)];q ˙i(t) = i[H, qi(t)]
❐ Because it is an important example let us look at the harmonic oscillator. The Hamiltonian is 1 H = (p2 + ω2q2) 2 0 Jorge C. Rom˜ao TCA-2012 – 8 Harmonic Oscillator
❐ The equations of motion are Lecture 1 Canonical Quantization 2 2 • Quantization in QM p˙ = i[H,p] = ω0q, q˙ = i[H, q] = p = q¨ + ω0q = 0 . • Quantization in FT − ⇒ • Symmetries ❐ It is convenient to introduce the operators Scalar fields
Lecture 2 1 † 1 Dirac field a = (ω0q + ip) ; a = (ω0q ip) Electromagnetic field √2ω0 √2ω0 − † Discrete Symmetries ❐ The equations of motion for a and a are very simple:
a˙(t) = iω a(t) e a˙ †(t) = iω a†(t) . − 0 0 They have the solution
−iω0t † † iω0t a(t) = a0e ; a (t) = a0e
❐ They obey the commutation relations
† † [a,a ]=[a0,a0] = 1 † † † † [a,a]=[a0,a0] = 0 , [a ,a ]=[a0,a0] = 0
Jorge C. Rom˜ao TCA-2012 – 9 Harmonic Oscillator
Lecture 1 ❐ In terms of a,a† the Hamiltonian reads Canonical Quantization • Quantization in QM • Quantization in FT 1 † †) 1 † † • Symmetries H = ω (a a + aa = ω (a a + a a ) 2 0 2 0 0 0 0 0 Scalar fields
Lecture 2 † 1 =ω0a0a0 + ω0 Dirac field 2 Electromagnetic field
Discrete Symmetries where we have used
[H,a ] = ω a , [H,a†] = ω a† 0 − 0 0 0 0 0
❐ † We see that a0 decreases the energy of a state by the quantity ω0 while a0 increases the energy by the same amount. ❐ As the Hamiltonian is a sum of squares the eigenvalues must be positive. Then it should exist a ground state (state with the lowest energy), 0 , defined by the condition | i
a 0 = 0 0 | i
Jorge C. Rom˜ao TCA-2012 – 10 Harmonic Oscillator
Lecture 1 n ❐ † Canonical Quantization The state n is obtained by the application of a0 . If we define • Quantization in QM | i • Quantization in FT • Symmetries 1 n n = a† 0 Scalar fields | i √n! 0 | i Lecture 2 Dirac field then Electromagnetic field Discrete Symmetries m n = δ h | i mn and 1 H n = n + ω n | i 2 0 | i
❐ † We will see that, in the quantum field theory, the equivalent of a0 and a0 are the creation and annihilation operators.
Jorge C. Rom˜ao TCA-2012 – 11 Canonical quantization for fields
❐ Let us move now to field theory, that is, systems with an infinite number of Lecture 1 degrees of freedom. To specify the state of the system, we must give for all Canonical Quantization • Quantization in QM space-time points one number (more if we are not dealing with a scalar field) • Quantization in FT • Symmetries ❐ The equivalent of the coordinates qi(t) and velocities, q˙i, are here the fields Scalar fields ϕ(~x, t) and their derivatives, ∂µϕ(~x, t). The action is now Lecture 2
Dirac field Electromagnetic field S = d4x (ϕ, ∂µϕ) Discrete Symmetries L Z where the Lagrangian density , is a functional of the fields ϕ and their derivatives ∂µϕ. L ❐ Let us consider closed systems for which does not depend explicitly on the coordinates xµ (energy and linear momentumL are therefore conserved). ❐ For simplicity let us consider systems described by n scalar fields ϕr(x),r = 1, 2, n. The stationarity of the action, δS = 0, implies the equations of motion,··· the so-called Euler-Lagrange equations, ∂ ∂ ∂µ L L = 0 r = 1, n ∂(∂µϕr) − ∂ϕr ···
Jorge C. Rom˜ao TCA-2012 – 12 Canonical quantization for fields
❐ For the case of real scalar fields with no interactions that we are considering, Lecture 1 we can easily see that the Lagrangian density should be, Canonical Quantization • Quantization in QM • Quantization in FT n • 1 µ 1 2 Symmetries = ∂ ϕr∂µϕr m ϕrϕr Scalar fields L 2 − 2 r=1 Lecture 2 X Dirac field in order to obtain the Klein-Gordon equations as the equations of motion, Electromagnetic field Discrete Symmetries ( + m2)ϕ =0 ; r = 1, n ⊔⊓ r ··· ❐ To define the canonical quantization rules we have to change to the Hamiltonian formalism, in particular we need to define the conjugate momentum π(x) for the field ϕ(x). ❐ To make an analogy with systems with n degrees of freedom, we divide the 3-dimensional space in cells with elementary volume ∆Vi. Then we introduce the coordinate ϕi(t) as the average of ϕ(~x, t) in the volume element ∆Vi, that is, 1 ϕ (t) d3xϕ(~x, t) i ≡ ∆V i Z(∆Vi) Jorge C. Rom˜ao TCA-2012 – 13 Canonical quantization for fields
❐ Also Lecture 1 1 3 Canonical Quantization ϕ˙ i(t) d xϕ˙(~x, t) . • Quantization in QM ≡ ∆Vi (∆Vi) • Quantization in FT Z • Symmetries ❐ Then
Scalar fields L = d3x ∆V . Lecture 2 L → iLi i Dirac field Z X Electromagnetic field ❐ Therefore the canonical momentum is now Discrete Symmetries ∂L ∂ i pi(t) = = ∆Vi L ∆Viπi(t) ∂ϕ˙ i(t) ∂ϕ˙ i(t) ≡ ❐ And the Hamiltonian
H = p ϕ˙ L = ∆V (π ϕ˙ ) i i − i i i − Li i i X X ❐ In the limit of the continuum, we define the conjugate momentum, ∂ (ϕ, ϕ˙) π(~x, t) L ≡ ∂ϕ˙(~x, t)
in such a way that its average value in ∆Vi is πi(t)
Jorge C. Rom˜ao TCA-2012 – 14 Canonical quantization for fields
Lecture 1 ❐ This suggests the introduction of an Hamiltonian density such that Canonical Quantization • Quantization in QM • Quantization in FT 3 • H = d x Symmetries H Scalar fields Z Lecture 2 =πϕ˙ . Dirac field H −L Electromagnetic field ❐ To define the rules of the canonical quantization we start with the Discrete Symmetries coordinates ϕi(t) and conjugate momenta pi(t). We have
[p (t),ϕ (t)] = iδ i j − ij
[ϕi(t),ϕj(t)]=0
[pi(t),pj(t)]=0
❐ In terms of momentum πi(t) we have
δij [πi(t),ϕj(t)] = i . − ∆Vi
Jorge C. Rom˜ao TCA-2012 – 15 Canonical quantization for fields
❐ Going into the continuum limit, ∆Vi 0, we obtain Lecture 1 → Canonical Quantization ′ ′ • Quantization in QM [ϕ(~x, t),ϕ(~x ,t)]=0 , [π(~x, t),π(~x ,t)]=0 • Quantization in FT ′ ′ • Symmetries [π(~x, t),ϕ(~x ,t)] = iδ(~x ~x ) Scalar fields − − Lecture 2 ❐ These relations are the basis of the canonical quantization. For the case of n Dirac field scalar fields, the generalization is: Electromagnetic field Discrete Symmetries ′ ′ [ϕr(~x, t),ϕs(~x ,t)]=0 [πr(~x, t),πs(~x ,t)]=0 [π (~x, t),ϕ (~x′,t)] = iδ δ(~x ~x′) r s − rs − where ∂ πr(~x, t) = L ∂ϕ˙ r(~x, t)
❐ The Hamiltonian is n H = d3x with = π ϕ˙ . H H r r −L r=1 Z X
Jorge C. Rom˜ao TCA-2012 – 16 Symmetries and conservation laws: Noether’s Theorem
❐ The Lagrangian formalism gives us a powerful method to relate symmetries Lecture 1 and conservation laws. At the classical level the fundamental result is Canonical Quantization • Quantization in QM • Quantization in FT ❐ Noether’s Theorem • Symmetries To each continuous symmetry transformation that leaves and the Scalar fields L Lecture 2 equations of motion invariant, corresponds one conservation law. Dirac field Electromagnetic field ❐ Instead of making the proof for all cases, we will consider three very Discrete Symmetries important particular cases: ❐ Translations Let us consider an infinitesimal translation x′µ = xµ + εµ
Then ∂ δ = ′ = εµ L L L −L ∂xµ
and ′ leads to the same equations of motion as , as they differ only by a 4-divergence.L If is invariant for translations, thenL it can not depend explicitly on theL coordinates xµ. Jorge C. Rom˜ao TCA-2012 – 17 Symmetries and conservation laws: Translations
❐ Therefore Lecture 1
Canonical Quantization ∂ ∂ ∂ ν • Quantization in QM δ = L δϕr + L δ(∂µϕr) = ∂µ L ε ∂ν ϕr • Quantization in FT L ∂ϕ ∂(∂ ϕ ) ∂(∂ ϕ ) r r µ r " r µ r # • Symmetries X X Scalar fields ν where we have used the equations of motion and δϕr = ε ∂ν ϕr. Lecture 2 µ Dirac field ❐ From the above and using the fact that ε is arbitrary we get
Electromagnetic field µν Discrete Symmetries ∂µT = 0 where T µν is the energy-momentum tensor defined by ∂ T µν = gµν + L ∂ν ϕ − L ∂(∂ ϕ ) r r µ r X ❐ Using these relations we can define the conserved quantities dP µ P µ d3xT 0µ = 0 ≡ ⇒ dt Z Noticing that T 00 = , it is easy to realize that P µ should be the 4-momentum vector.H Therefore we conclude that invariance for translations leads to the conservation of energy and momentum. Jorge C. Rom˜ao TCA-2012 – 18 Symmetries and conservation laws: Lorentz transformations
❐ Consider the infinitesimal Lorentz transformations Lecture 1 Canonical Quantization ′µ µ µ ν • Quantization in QM x = x + ω ν x • Quantization in FT • Symmetries ❐ The transformation for the coordinates will induce the following Scalar fields transformation for the fields, Lecture 2
Dirac field ′ ′ ϕr(x ) = Srs(ω)ϕs(x) Electromagnetic field ❐ Discrete Symmetries For the case of scalar fields Srs = δrs and for spinors we know that 1 µν Srs = δrs + 8 [γµ,γν ]rsω . In general the variation of ϕr comes from two different effects ❐ We have
δϕ (x) ϕ′ (x) ϕ (x) = S−1(ω)ϕ (x′) ϕ (x) r ≡ r − r rs s − r 1 = ω (xα∂β xβ∂α)δ +Σαβ ϕ − 2 αβ − rs rs s where we have defined 1 S (ω) = δ + ω Σαβ . rs rs 2 αβ rs Jorge C. Rom˜ao TCA-2012 – 19 Symmetries and conservation laws: Lorentz transformations
❐ Then Lecture 1 Canonical Quantization ∂ • Quantization in QM δ = ∂µ L δϕr • Quantization in FT L ∂(∂µϕr) • Symmetries Scalar fields which gives Lecture 2
Dirac field µαβ µαβ α µβ β µα ∂ αβ Electromagnetic field ∂µM = 0 with M = x T x T + L Σrs ϕs − ∂(∂µϕr) Discrete Symmetries
❐ The conserved angular momentum is then
M αβ = d3xM 0αβ = d3x xαT 0β xβT 0α + π Σαβϕ − r rs s " r,s # Z Z X with dM αβ = 0 dt
Jorge C. Rom˜ao TCA-2012 – 20 Symmetries and conservation laws: Internal symmetries
Lecture 1 ❐ Let us consider that the Lagrangian is invariant for an infinitesimal internal Canonical Quantization symmetry transformation • Quantization in QM • Quantization in FT • Symmetries δϕr(x) = iελrsϕs(x) Scalar fields − Lecture 2 ❐ Dirac field Then we can easily show that
Electromagnetic field
Discrete Symmetries µ µ ∂ ∂µJ = 0 where J = i L λrsϕs − ∂(∂µϕr)
❐ This leads to the conserved charge dQ Q(λ) = i d3xπ λ ϕ ; = 0 − r rs s dt Z
Jorge C. Rom˜ao TCA-2012 – 21 Symmetries and conservation laws: Quantum Theory
❐ These relations between symmetries and conservation laws were derived for Lecture 1 the classical theory Canonical Quantization • Quantization in QM ❐ In the quantum theory the fields ϕr(x) become operators acting on the • Quantization in FT • Symmetries Hilbert space of the states. The physical observables are related with the Scalar fields matrix elements of these operators. We have therefore to require Lorentz Lecture 2 covariance for those matrix elements Dirac field ❐ This means that the classical fields relation Electromagnetic field
Discrete Symmetries ′ ′ ϕr(x ) = Srs(a)ϕs(x)
should be in the quantum theory
Φ′ ϕ (x′) Φ′ = S (a) Φ ϕ (x) Φ α| r | β rs h α| s | βi ❐ There should exist an unitary transformation U(a, b) that should relate the two inertial frames
Φ′ = U(a, b) Φ | i | i µ µ where a ν e b are defined by ′µ µ ν µ x = a ν x + b Jorge C. Rom˜ao TCA-2012 – 22 Symmetries and conservation laws: Quantum Theory
❐ Therefore we get that the field operators should transform as Lecture 1 Canonical Quantization −1 (−1) • Quantization in QM U(a, b)ϕr(x)U (a, b) = Srs (a)ϕs(ax + b) • Quantization in FT • Symmetries ❐ Let us look at the consequences of this relation for translations and Lorentz Scalar fields transformations. We consider first the translations. We get Lecture 2
Dirac field −1 Electromagnetic field U(b)ϕr(x)U (b) = ϕr(x + b)
Discrete Symmetries ❐ For infinitesimal translations we can write
µ U(ε) eiεµP 1 + iε µ ≡ ≃ µP where µ is an hermitian operator. P ❐ This gives
i[ µ,ϕ (x)] = ∂µϕ (x) P r r ❐ The correspondence with classical mechanics and non relativistic quantum theory suggests that we identify µ with the 4-momentum, that is, µ P µ where P µ has been definedP before P ≡ Jorge C. Rom˜ao TCA-2012 – 23 Symmetries and conservation laws: Quantum Theory
′µ µ ν ❐ For Lorentz transformations x = a ν x , we write for an infinitesimal Lecture 1 transformation Canonical Quantization • Quantization in QM • Quantization in FT aµ = gµ + ωµ + O(ω2) • Symmetries ν ν ν
Scalar fields
Lecture 2 and therefore
Dirac field i µν Electromagnetic field U(ω) 1 ωµν Discrete Symmetries ≡ − 2 M ❐ We then obtain the requirement
i[ µν ,ϕ (x)] = xµ∂ν ϕ xν ∂µϕ +Σµν ϕ (x) M r r − r rs s ❐ As we have an explicit expression for P µ and M µν and we know the commutation relations of the quantum theory, these equations become an additional requirement that the theory has to verify in order to be invariant under translations. We will see explicitly that this is indeed the case for the theories in which we are interested.
Jorge C. Rom˜ao TCA-2012 – 24 Real scalar field
❐ The real scalar field described by the Lagrangian density Lecture 1
Canonical Quantization 1 µ 1 2 Scalar fields = ∂ ϕ∂µϕ m ϕϕ • Real scalar field L 2 − 2 • Causality • Vac fluctuations to which corresponds the Klein-Gordon equation • Charged scalar field • Feynman Propagator ( + m2)ϕ = 0 Lecture 2 ⊔⊓ Dirac field Electromagnetic field is the simplest example, and in fact was already used to introduce the Discrete Symmetries general formalism. ❐ As we have seen the conjugate momentum is ∂ π = L =ϕ ˙ ∂ϕ˙
❐ The commutation relations are
[ϕ(~x, t),ϕ(~x′,t)]=[π(~x, t),π(~x′,t)]=0
[π(~x, t),ϕ(~x′,t)] = iδ3(~x ~x′) − − Jorge C. Rom˜ao TCA-2012 – 25 Real scalar field
❐ The Hamiltonian is given by, Lecture 1
Canonical Quantization 0 3 3 1 2 1 2 1 2 2 Scalar fields H = P = d x = d x π + ~ ϕ + m ϕ • Real scalar field H 2 2|∇ | 2 • Causality Z Z • Vac fluctuations • Charged scalar field and the linear momentum is • Feynman Propagator Lecture 2 P~ = d3xπ~ ϕ Dirac field − ∇ Electromagnetic field Z Discrete Symmetries ❐ Using these equations it is easy to verify that
i[P µ,ϕ] = ∂µϕ
showing the invariance of the theory for the translations. In the same way we µν can verify the invariance under Lorentz transformations, with Σrs = 0 (spin zero). ❐ In order to define the states of the theory it is convenient to have eigenstates of energy and momentum.
Jorge C. Rom˜ao TCA-2012 – 26 Real scalar field
❐ To build these states we start by making a spectral Fourier decomposition of Lecture 1 ϕ(~x, t) in plane waves: Canonical Quantization Scalar fields −ik·x † ik·x • Real scalar field ϕ(~x, t) = dk a(k)e + a (k)e • Causality Z • Vac fluctuations where • Charged scalar field f • Feynman Propagator d3k ~ 2 2 Lecture 2 dk 3 ; ωk = + k + m ≡ (2π) 2ωk | | Dirac field q Electromagnetic field is thef Lorentz invariant integration measure. Discrete Symmetries ❐ As in the quantum theory ϕ is an operator, also a(k) e a†(k) should be operators. As ϕ is real, then a†(k) should be the hermitian conjugate to a(k). In order to determine their commutation relations we start by solving in order to a(k) and a†(k). Using the properties of the delta function, we get
↔ ↔ a(k) = i d3xeik·x∂ ϕ(x) , a†(k) = i d3xe−ik·x∂ ϕ(x) 0 − 0 where we haveZ introduced the notation Z ↔ ∂b ∂a a∂ b = a b 0 ∂t − ∂t Jorge C. Rom˜ao TCA-2012 – 27 Real scalar field
❐ a(k) and a†(k) time independent as can be checked explicitly (see the Lecture 1
Canonical Quantization Problems). This observation is important in order to be able to choose equal
Scalar fields times in the commutation relations. We get • Real scalar field • Causality ↔ ′ ↔ • Vac fluctuations † ′ 3 3 ik·x −ik ·y [a(k),a (k )] = d x d y e ∂ 0ϕ(~x, t), e ∂ 0ϕ(~y, t) • Charged scalar field • Feynman Propagator Z Z h i Lecture 2 3 3 ′ =(2π) 2ωkδ (~k ~k ) Dirac field − Electromagnetic field and Discrete Symmetries [a(k),a(k′)]=[a†(k),a†(k′)]=0
❐ We then see that, except for a small difference in the normalization, a(k) e a†(k) should be interpreted as annihilation and creation operators of states with momentum kµ. To show this, we observe that 1 H = dk ω a†(k)a(k) + a(k)a†(k) 2 k Z 1 f P~ = dk ~k a†(k)a(k) + a(k)a†(k) 2 Z Jorge C. Rom˜ao f TCA-2012 – 28 Real scalar field
❐ Using these explicit forms we can then obtain Lecture 1 Canonical Quantization [P µ,a†(k)] = kµa†(k) [P µ,a(k)] = kµa(k) Scalar fields − • Real scalar field † µ • Causality showing that a (k) adds momentum k and that a(k) destroys momentum • Vac fluctuations µ • k . That the quantization procedure has produced an infinity number of Charged scalar field † • Feynman Propagator oscillators should come as no surprise. In fact a(k),a (k) correspond to the Lecture 2 quantization of the normal modes of the classical Klein-Gordon field. Dirac field Electromagnetic field ❐ By analogy with the harmonic oscillator, we are now in position of finding Discrete Symmetries the eigenstates of H. We start by defining the base state, that in quantum field theory is called the vacuum. We have
a(k) 0 =0 ; | ik ∀k ❐ Then the vacuum, that we will denote by 0 , will be formally given by | i 0 =Π 0 | i k | ik and we will assume that it is normalized, that is 0 0 = 1. If now we calculate the vacuum energy, we find immediatelyh the| i first problem with infinities in Quantum Field Theory (QFT).
Jorge C. Rom˜ao TCA-2012 – 29 Real scalar field
❐ In fact Lecture 1 Canonical Quantization 1 0 H 0 = dk ω 0 a†(k)a(k) + a(k)a†(k) 0 Scalar fields h | | i 2 k | | • Real scalar field Z • Causality • Vac fluctuations 1 f † • Charged scalar field = dk ωk 0 a(k),a (k) 0 • Feynman Propagator 2 | | Z Lecture 2 f 3 Dirac field 1 d k 3 3 = 3 ωk(2π) 2ωkδ (0) Electromagnetic field 2 (2π) 2ωk Discrete Symmetries Z 1 = d3k ω δ3(0) = 2 k ∞ Z ❐ This infinity can be understood as the the (infinite) sum of the zero point energy of all quantum oscillators. In the discrete case we would have, 1 k 2 ωk = . This infinity can be easily removed. We start by noticing that we only∞ measure energies as differences with respect to the vacuum Penergy, and those will be finite. We will then define the energy of the vacuum as being zero. Technically this is done as follows.
Jorge C. Rom˜ao TCA-2012 – 30 Real scalar field
❐ We define a new operator P µ as Lecture 1 N.O.
Canonical Quantization µ 1 µ † † Scalar fields PN.O. dk k a (k)a(k) + a(k)a (k) • Real scalar field ≡2 • Causality Z • Vac fluctuations 1 µ † † • f dk k 0 a (k)a(k) + a(k)a (k) 0 Charged scalar field − 2 | | • Feynman Propagator Z Lecture 2 = dk kµfa†(k)a(k) Dirac field Electromagnetic field ❐ µ Z Now 0 PN.O. 0 = 0. The ordering of operators where the annihilation Discrete Symmetries f operatorsh | appear| i on the right of the creation operators is called normal ordering and the usual notation is 1 : (a†(k)a(k) + a(k)a†(k)) : a†(k)a(k) 2 ≡ ❐ To remove the infinity of the energy and momentum corresponds to choose the normal ordering. We will adopt this convention in the following dropping the subscript ”N.O.” to simplify the notation. This should not appear as an ad hoc procedure. In fact, in going from the classical theory where we have products of fields into the quantum theory where the fields are operators, we should have a prescription for the correct ordering of such products.
Jorge C. Rom˜ao TCA-2012 – 31 Real scalar field
❐ Once we have the vacuum we can build the states by applying the creation Lecture 1 operators a†(k). As in the case of the harmonic oscillator, we can define the Canonical Quantization number Scalar fields operator, • Real scalar field • Causality † • Vac fluctuations N = dk a (k)a(k) • Charged scalar field • Feynman Propagator Z ❐ It is easy to see that N commutes with H and therefore the eigenstates of Lecture 2 f Dirac field H are also eigenstates of N.
Electromagnetic field ❐ The state with one particle of momentum kµ is obtained as a†(k) 0 . In fact Discrete Symmetries we have | i
′ P µa†(k) 0 = dk k′µa†(k′)a(k′)a†(k) 0 | i | i Z = df3k′k′µδ3(~k ~k′)a†(k) 0 − | i Z =kµa†(k) 0 | i and
Na†(k) 0 = a†(k) 0 | i | i Jorge C. Rom˜ao TCA-2012 – 32 Real scalar field
† † ❐ In a similar way, the state a (k1)...a (kn) 0 would be a state with n Lecture 1 | i Canonical Quantization particles. However, the sates that we have just defined have a problem.
Scalar fields They are not normalizable and therefore they can not form a basis for the • Real scalar field Hilbert space of the quantum field theory, the so-called Fock space. • Causality • Vac fluctuations • Charged scalar field ❐ The origin of the problem is related to the use of plane waves and states • Feynman Propagator with exact momentum. This can be solved forming states that are Lecture 2
Dirac field superpositions of plane waves
Electromagnetic field † Discrete Symmetries 1 = λ dk C(k)a (k) 0 | i | i Z ❐ Then f 1 1 =λ2 dk dk C∗(k )C(k ) 0 a(k )a†(k ) 0 h | i 1 2 1 2 | 1 2 | Z
=λ2 dkf Cf(k) 2 = 1 | | and therefore Z f −1/2 λ = dk C(k) 2 if dk C(k) 2 < | | | | ∞ Z Z Jorge C. Rom˜ao f f TCA-2012 – 33 Real scalar field
❐ If k is only different from zero in a neighborhood of a given 4-momentum Lecture 1 kµ, then the state will have a well defined momentum (within some Canonical Quantization
Scalar fields experimental error). • Real scalar field ❐ A basis for the Fock space can then be constructed from the n–particle • Causality • Vac fluctuations normalized states • Charged scalar field • Feynman Propagator −1/2 2 Lecture 2 n = n! dk dk C(k , k ) | i 1 ··· n| 1 ··· n | Dirac field Z Electromagnetic field dk f dk Cf(k , k )a†(k ) a†(k ) 0 Discrete Symmetries 1 ··· n 1 ··· n 1 ··· n | i Z that satisfy n fn = 1f, N n = n n h | i | i | i ❐ Due to the commutation relations of the operators a†(k), the functions C(k k ) are symmetric (obey the Bose–Einstein statistics), 1 ··· n C( k , k , ) = C( k k ) ··· i ··· j ··· ··· j ··· i ··· ❐ This interpretation in terms of particles, with creation and annihilation operators, that results from the canonical quantization, is usually called second quantization, as opposed to the description in terms of wave functions (the first quantization) Jorge C. Rom˜ao TCA-2012 – 34 Microscopic causality
❐ Classically, the fields can be measured with an arbitrary precision. In a Lecture 1 relativistic quantum theory we have several problems. The first, results from Canonical Quantization
Scalar fields the fact that the fields are now operators. This means that the observables • Real scalar field should be connected with the matrix elements of the operators and not with • Causality • Vac fluctuations the operators. • Charged scalar field • Feynman Propagator ❐ Besides this question, we can only speak of measuring ϕ in two space-time Lecture 2 points x and y if [ϕ(x),ϕ(y)] vanishes. Let us look at the conditions needed Dirac field
Electromagnetic field for this to occur.
Discrete Symmetries † −ik1·x+ik2·y † [ϕ(x),ϕ(y)] = dk1dk2 a(k1),a (k2) e + a (k1),a(k2) e Z = dkf fe−ik1·(x−y) eik1·(x−y) 1 − Z i∆(fx y) ≡ − ❐ The function ∆(x y) is Lorentz invariant and satisfies the relations − ( + m2)∆(x y) = 0 ⊔⊓x − ∆(x y) = ∆(y x) ∆(~x ~y, 0) = 0 − − − − Jorge C. Rom˜ao TCA-2012 – 35 Microscopic causality
Lecture 1 ❐ The last relation ensures that the equal time commutator of two fields Canonical Quantization vanishes. Lorentz invariance implies then, Scalar fields • Real scalar field • Causality ∆(x y)=0 ; (x y)2 < 0 • Vac fluctuations − ∀ − • Charged scalar field • Feynman Propagator ❐ This means that for two points that can not be physically connected, that is Lecture 2 for which (x y)2 < 0, the fields interpreted as physical observables, can Dirac field − Electromagnetic field then be independently measured. This result is known as Microscopic Discrete Symmetries Causality. ❐ We also note that
0 3 ∂ ∆(x y) 0 0 = δ (~x ~y) − |x =y − − which ensures the canonical commutation relation
Jorge C. Rom˜ao TCA-2012 – 36 Vacuum fluctuations
❐ It is well known from Quantum Mechanics that, in an harmonic oscillator, Lecture 1 the coordinate is not well defined for the energy eigenstates, that is Canonical Quantization Scalar fields 2 2 • Real scalar field n q n > ( n q n ) = 0 • Causality | | h | | i • Vac fluctuations • Charged scalar field ❐ In Quantum Field Theory, we deal with an infinite set of oscillators, and • Feynman Propagator
Lecture 2 therefore we will have the same behavior, that is,
Dirac field 0 ϕ(x)ϕ(y) 0 = 0 Electromagnetic field h | | i 6 Discrete Symmetries although
0 ϕ(x) 0 = 0 h | | i ❐ We can calculate the above expression. We have
0 ϕ(x)ϕ(y) 0 = dk dk e−ik1·xeik2·y 0 a(k )a†(k ) 0 h | | i 1 2 | 1 2 | Z
f f = dk e−ik·(x−y) ∆ (x y) 1 ≡ + − Z Jorge C. Rom˜ao f TCA-2012 – 37 Vacuum fluctuations
❐ The function ∆+(x y) corresponds to the positive frequency part of Lecture 1 ∆(x y). When y − x this expression diverges quadratically, Canonical Quantization − → Scalar fields • Real scalar field • Causality 3 • d k Vac fluctuations 0 ϕ2(x) 0 = ∆ (0) = dk = 1 • Charged scalar field | | + 1 (2π)32ω • Feynman Propagator Z Z k1 Lecture 2 f Dirac field ❐ This divergence can not be eliminated in the way we did with the energy of Electromagnetic field the vacuum. In fact these vacuum fluctuations, as they are known, do have Discrete Symmetries observable consequences like, for instance, the Lamb shift.
❐ We will be less worried with this result, if we notice that for measuring the square of the operator ϕ at x we need frequencies arbitrarily large, that is, an infinite amount of energy. Physically only averages over a finite space-time region have meaning.
Jorge C. Rom˜ao TCA-2012 – 38 Charged scalar field
❐ The description in terms of real fields does not allow the distinction between Lecture 1 particles and anti-particles. It applies only the those cases were the particle Canonical Quantization 0 Scalar fields and anti-particle are identical, like the π . For the more usual case where • Real scalar field particles and anti-particles are distinct, it is necessary to have some charge • Causality • Vac fluctuations (electric or other) that allows us to distinguish them. For this we need • Charged scalar field complex fields. • Feynman Propagator Lecture 2 ❐ The theory for the scalar complex field can be easily obtained from two real Dirac field
Electromagnetic field scalar fields ϕ1 and ϕ2 with the same mass. If we denote the complex field ϕ
Discrete Symmetries by, ϕ + iϕ ϕ = 1 2 √2 then
= (ϕ ) + (ϕ ) =: ∂µϕ†∂ ϕ m2ϕ†ϕ : L L 1 L 2 µ − which leads to the equations of motion
( + m2)ϕ =0; ( + m2)ϕ† = 0 ⊔⊓ ⊔⊓ Jorge C. Rom˜ao TCA-2012 – 39 Charged scalar field
❐ The classical theory has, at the classical level, a conserved current, Lecture 1 µ ∂µJ = 0, Canonical Quantization Scalar fields ↔µ • Real scalar field µ † • Causality J = iϕ ∂ ϕ • Vac fluctuations • Charged scalar field • Feynman Propagator ❐ Therefore we expect, at the quantum level, the charge Q Lecture 2 Dirac field Q = d3x : i(ϕ†ϕ˙ ϕ˙ †ϕ): Electromagnetic field − Discrete Symmetries Z to be conserved, that is, [H, Q] = 0. ❐ To show this we need to know the commutation relations for the field ϕ. The definition and the commutation relations for ϕ1 and ϕ2 allow us to obtain the following relations for ϕ and ϕ†:
[ϕ(x),ϕ(y)]=[ϕ†(x),ϕ†(y)]=0 [ϕ(x),ϕ†(y)] = i∆(x y) −
Jorge C. Rom˜ao TCA-2012 – 40 Charged scalar field
❐ For equal times we can get Lecture 1 Canonical Quantization [π(~x, t),ϕ(~y, t)]=[π†(~x, t),ϕ†(~y, t)] = iδ3(~x ~y) Scalar fields − − • Real scalar field • Causality • Vac fluctuations where • Charged scalar field • Feynman Propagator π =ϕ ˙ † ; π† =ϕ ˙ Lecture 2
Dirac field Electromagnetic field ❐ The plane waves expansion is then Discrete Symmetries −ik·x † ik·x ϕ(x) = dk a+(k)e + a−(k)e Z h i † f −ik·x † ik·x ϕ (x) = dk a−(k)e + a+(k)e Z h i where the definitionf of a±(k) is
† † a1(k) ia2(k) † a1(k) ia2(k) a±(k) = ± ; a = ∓ √2 ± √2
Jorge C. Rom˜ao TCA-2012 – 41 Charged scalar field
❐ The algebra of the operators a± it is easily obtained from the algebra of the Lecture 1 operators ai′ s. We get the following non-vanishing commutators: Canonical Quantization Scalar fields † ′ † ′ 3 3 ~ ~ ′ • Real scalar field [a+(k),a+(k )]=[a−(k),a−(k )] = (2π) 2ωkδ (k k ) • Causality − • Vac fluctuations • Charged scalar field † • Feynman Propagator therefore allowing us to interpret a+ and a+ as annihilation and creation Lecture 2 operators of quanta of type +, and similarly for the quanta of type . − Dirac field ❐ Electromagnetic field We can construct the number operators for those quanta:
Discrete Symmetries † N± = dk a±(k)a±(k) Z f ❐ One can easily verify that
N+ + N− = N1 + N2
where
† Ni = dk ai (k)ai(k) Z Jorge C. Rom˜ao f TCA-2012 – 42 Charged scalar field
❐ The energy-momentum operator can be written in terms of the + and Lecture 1 operators, − Canonical Quantization
Scalar fields • Real scalar field µ µ † † • Causality P = dk k a+(k)a+(k) + a−(k)a−(k) • Vac fluctuations Z • Charged scalar field h i • Feynman Propagator where we havef already considered the normal ordering. Lecture 2 Dirac field ❐ We also obtain for the charge Q: Electromagnetic field Discrete Symmetries Q = d3x : i(ϕ†ϕ˙ ϕ˙ †ϕ˙): − Z † † = dk a (k)a (k) a (k)a−(k) + + − − Z h i =N N− +f− ❐ Using the commutation relations one can easily verify that
[H, Q] = 0
showing that the charge Q is conserved. The previous equation allows us to interpret the quanta as having charge 1. ± ± Jorge C. Rom˜ao TCA-2012 – 43 Charged scalar field
Lecture 1 ❐ However, before introducing interactions, the theory is symmetric, and we Canonical Quantization can not distinguish between the two types of quanta. Scalar fields • Real scalar field • Causality ❐ From the commutation relations we obtain, • Vac fluctuations • Charged scalar field µ † µ † • Feynman Propagator [P ,a+(k)] = k a+(k) Lecture 2 † † Dirac field [Q, a+(k)]=+a+(k) Electromagnetic field
Discrete Symmetries ❐ † µ This shows that a+(k) creates a quanta with 4-momentum k and charge +1. In a similar way we can show that a† creates a quanta with charge 1 − − and that a±(k) annihilate quanta of charge 1, respectively. ±
Jorge C. Rom˜ao TCA-2012 – 44 Time ordered product and the Feynman propagator
❐ The operator ϕ† creates a particle with charge +1 or annihilates a particle Lecture 1 with charge 1. In both cases it adds a total charge +1. In a similar way ϕ Canonical Quantization − Scalar fields annihilates one unit of charge. Let us construct a state of one particle (not • Real scalar field normalized) with charge +1 by application of ϕ† in the vacuum: • Causality • Vac fluctuations • Charged scalar field † Ψ+(~x, t) ϕ (~x, t) 0 • Feynman Propagator | i ≡ | i Lecture 2 ❐ The amplitude to propagate the state Ψ+ into the future to the point Dirac field (~x′,t′) with t′ >t is given by | i Electromagnetic field
Discrete Symmetries θ(t′ t) Ψ (~x′,t′) Ψ (~x, t) = θ(t′ t) 0 ϕ(~x′,t′)ϕ†(~x, t) 0 − h + | + i − | | ❐ In ϕ†(~x, t) 0 only the operator a† (k) is active, while in 0 ϕ(~x′,t′) the | i + h | same happens to a+(k). Therefore this is the matrix element that creates a quanta of charge +1 in (~x, t) and annihilates it in (~x′,t′) with t′ >t.
❐ There exists another way of increasing the charge by +1 unit in (~x, t) and decreasing it by 1 in (~x′,t′). This is achieved if we create a quanta of charge 1 in ~x′ −at time t′ and let it propagate to ~x where it is absorbed at time t>t− ′. The amplitude is then, ′ ′ ′ † ′ ′ ′ θ(t t ) Ψ−(~x, t) Ψ−(~x ,t ) = 0 ϕ (~x, t)ϕ(~x ,t ) 0 θ(t t ) − h | i | | − Jorge C. Rom˜ao TCA-2012 – 45 Time ordered product and the Feynman propagator
❐ Since we can not distinguish the two paths we must sum of the two Lecture 1 amplitudes. This is the so-called Feynman propagator. It can be written in a Canonical Quantization
Scalar fields more compact way if we introduce the time ordered product. Given two ′ • Real scalar field operators a(x) and b(x ) we define the time ordered product T by, • Causality • Vac fluctuations ′ ′ ′ ′ ′ • Charged scalar field Ta(x)b(x ) θ(t t )a(x)b(x ) + θ(t t)b(x )a(x) • Feynman Propagator ≡ − − Lecture 2 ❐ In this prescription the older times are always to the right of the more recent Dirac field times. It can be applied to an arbitrary number of operators. With this Electromagnetic field definition, the Feynman propagator reads, Discrete Symmetries ∆ (x′ x) = 0 Tϕ(x′)ϕ†(x) 0 F − | | † ❐ Using the ϕ and ϕ decomposition we can calculate ∆F (for free fields)
′ ′ ∆ (x′ x) = dk θ(t′ t)e−ik·(x −x) + θ(t t′)eik·(x −x) F − − − Z 4 h i d k i ′ = f e−ik·(x −x) (2π)4 k2 m2 + iε Z 4 − d k ′ ∆ (k)e−ik·(x −x) ≡ (2π)4 F Z Jorge C. Rom˜ao TCA-2012 – 46 Time ordered product and the Feynman propagator
i Lecture 1 ❐ Where ∆ (k) F k2 m2 + iε Canonical Quantization ≡ ❐ − Scalar fields ∆F (k) is the propagator in momenta space (Fourier transform). The • Real scalar field equivalence between the previous equations is done using integration in the • Causality 0 • Vac fluctuations complex plane of the time component k , with the help of the residue • Charged scalar field • Feynman Propagator theorem. The contour is defined by the iε prescription, as indicated in in the figure Lecture 2 Im k0 Dirac field
Electromagnetic field
Discrete Symmetries
Re k0
❐ Applying the operator ( ′ + m2) to ∆ (x′ x) one can show that ⊔⊓x F − ( ′ + m2)∆ (x′ x) = iδ4(x′ x) ⊔⊓x F − − − ′ that is, ∆F (x x) is the Green’s function for the Klein-Gordon equation with Feynman boundary− conditions.
Jorge C. Rom˜ao TCA-2012 – 47 Lecture 1
Canonical Quantization
Scalar fields
Lecture 2
Dirac field
Electromagnetic field
Discrete Symmetries Lecture 2
Jorge C. Rom˜ao TCA-2012 – 48 Canonical formalism for the Dirac field
❐ Let us now apply the formalism of second quantization to the Dirac field. Lecture 1 Something has to be changed, otherwise we would be led to a theory Canonical Quantization
Scalar fields obeying Bose statistics, while we know that electrons obey Fermi statistics. Lecture 2 ❐ The Lagrangian density that leads to the Dirac equation is Dirac field • Quantization • Causality µ = iψγ ∂µψ mψψ • Feynman propagator L − Electromagnetic field
Discrete Symmetries The conjugate momentum to ψα is
∂ † πα = L = iψα ∂ψ˙α
† while the conjugate momentum to ψα vanishes. The Hamiltonian density is
= πψ˙ = ψ†( i~α ~ + βm)ψ H −L − · ∇ ❐ The requirement of translational and Lorentz invariance for leads to the tensors T µν and M µνλ. We get for energy-momentum tensorL
T µν = iψγµ∂ν ψ gµν − L Jorge C. Rom˜ao TCA-2012 – 49 Canonical formalism for the Dirac field
❐ And for the angular-momentum tensor density Lecture 1 Canonical Quantization M µνλ = iψγµ(xν∂λ xλ∂ν + σνλ)ψ xν gµλ xµgνλ Scalar fields − − − L Lecture 2 where Dirac field • Quantization • Causality νλ 1 ν λ • σ = [γ ,γ ] Feynman propagator 4 Electromagnetic field
Discrete Symmetries ❐ The 4-momentum P µ and the angular momentum tensor M νλ are then given by,
P µ d3xT 0µ, M µλ d3xM 0νλ ≡ ≡ Z Z ❐ This gives for the energy and linear momentum
H d3xψ†( i~α ~ + βm)ψ ≡ − · ∇ Z P~ d3xψ†( i~ )ψ ≡ − ∇ Z Jorge C. Rom˜ao TCA-2012 – 50 Canonical formalism for the Dirac field
❐ If we define the angular momentum vector J~ (M 23, M 31, M 12) we get Lecture 1 ≡ Canonical Quantization 1 1 Scalar fields J~ = d3xψ† ~r ~ + Σ~ ψ Lecture 2 × i ∇ 2 Z Dirac field • Quantization which has the familiar aspect J~ = L~ + S~. • Causality • Feynman propagator µ µ µ ❐ We can also identify a conserved current, ∂µj = 0, with j = ψγ ψ, which Electromagnetic field will give the conserved charge Discrete Symmetries
Q = d3xψ†ψ Z ❐ All that we have done so far is at the classical level. To apply the canonical formalism we have to enforce commutation relations and verify the Lorentz invariance of the theory. This will lead us into problems. To see what these are and how to solve them, we will introduce the plane wave expansions,
ψ(x) = dp b(p,s)u(p,s)e−ip·x + d†(p,s)v(p,s)eip·x Z s X f ψ†(x) = dp b†(p,s)u†(p,s)e+ip·x + d(p,s)v†(p,s)e−ip·x Z s X Jorge C. Rom˜ao f TCA-2012 – 51 Canonical formalism for the Dirac field
❐ u(p,s) and v(p,s) are the spinors for positive and negative energy, Lecture 1 respectively, introduced in the study of the Dirac equation and b, b†, d and d† Canonical Quantization
Scalar fields are operators. Lecture 2 ❐ Dirac field To see what are the problems with the canonical quantization of fermions, µ • Quantization let us calculate P . We get • Causality • Feynman propagator † † Electromagnetic field P µ = dk kµ b (k,s)b(k,s) d(k,s)d (k,s) Discrete Symmetries − Z s X f where we have used the orthogonality and closure relations for the spinors. ❐ We realize that if we define the vacuum as b(k,s) 0 = d(k,s) 0 = 0 and if we quantize with commutators then particles b and| particlesi d will| i contribute with opposite signs to the energy and the theory will not have a stable ground state. ❐ In fact, this was the problem already encountered in the study of the negative energy solutions of the Dirac equation, and this is the reason for the negative sign. Dirac’s hole theory required Fermi statistics for the electrons and we will see how spin and statistics are related.
Jorge C. Rom˜ao TCA-2012 – 52 Canonical formalism for the Dirac field
❐ To discover what are the relations that b, b†, d and d† should obey, we recall Lecture 1
Canonical Quantization that at the quantum level it is always necessary to verify Lorentz invariance.
Scalar fields
Lecture 2 i[Pµ, ψ(x)] = ∂µψ ; i[Pµ, ψ(x)] = ∂µψ
Dirac field • Quantization ❐ • Causality We start with the above equations and we will discover the appropriate • Feynman propagator relations for the operators. Using the plane wave expansions we can show Electromagnetic field that the above equations lead to Discrete Symmetries [P , b(k,s)] = k b(k,s); [P , b†(k,s)] = k b†(k,s) µ − µ µ µ [P , d(k,s)] = k d(k,s); [P , d†(k,s)] = k d†(k,s) µ − µ µ µ
❐ From the definition of Pµ we get
b†(p,s′)b(p,s′) d(p,s′)d†(p,s′) , b(k,s) = (2π)32k0δ3(~k ~p)b(k,s) − − − s′ X and three other similar relations.
Jorge C. Rom˜ao TCA-2012 – 53 Canonical formalism for the Dirac field
❐ If we assume that Lecture 1 Canonical Quantization [d†(p,s′)d(p,s′), b(k,s)]=0 Scalar fields Lecture 2 the previous condition reads Dirac field • Quantization • Causality b†(p,s′) b(p,s′), b(k,s) b†(p,s′), b(k,s) b(p,s′) = • Feynman propagator { } − { } s′ Electromagnetic field X Discrete Symmetries = (2π)32k0δ3(~p ~k)b(k,s) − − where the parenthesis , denote anti-commutators. { } ❐ It is easy to see that this relation is verified if we impose the canonical anti-commutation relations. We should have
† 3 0 3 b (p,s), b(k,s) = (2π) 2k δ (~p ~k)δ ′ { } − ss † ′ 3 0 3 d (p,s ), d(k,s) = (2π) 2k δ (~p ~k)δ ′ { } − ss and all the other anti-commutators vanish. Note that as b anti-commutes with d and d†, then it commutes with d†d as we have assumed.
Jorge C. Rom˜ao TCA-2012 – 54 Canonical formalism for the Dirac field
❐ With the anti-commutator relations both contributions to P µ are positive. Lecture 1 As in boson case we have to subtract the zero point energy. This is done, as Canonical Quantization µ Scalar fields usual, by taking all quantities normal ordered. Therefore we have for P ,
Lecture 2 † † Dirac field P µ = dk kµ : b (k,s)b(k,s) d(k,s)d (k,s) : • Quantization − • Causality Z s • Feynman propagator X f Electromagnetic field = dk kµ : b†(k,s)b(k,s) + d†(k,s)d(k,s) : Discrete Symmetries Z s X ❐ For the charge f
Q = d3x : ψ†(x)ψ(x) := dk b†(k,s)b(k,s) d†(k,s)d(k,s) − Z Z s X f which means that the quanta of b type have charge +1 while those of d type have charge 1. It is interesting to note that was the second quantization of the Dirac field− that introduced the sign in the charge, making the charge operator without a definite sign, while− in Dirac theory was the probability density that was positive defined. The reverse is true for bosons.
Jorge C. Rom˜ao TCA-2012 – 55 Canonical formalism for the Dirac field
❐ We can easily show that Lecture 1 [Q, b†(k,s)] = b†(k,s) , [Q, d(k,s)] = d(k,s), [Q, ψ] = ψ Canonical Quantization − Scalar fields [Q, b(k,s)] = b(k,s) , [Q, d†(k,s)] = d†(k,s) [Q, ψ] = ψ Lecture 2 − − Dirac field ❐ In QED the charge is given by eQ (e < 0). Therefore we see that ψ creates • Quantization • Causality positrons and annihilates electrons and the opposite happens with ψ. • Feynman propagator ❐ We can introduce the number operators Electromagnetic field
Discrete Symmetries N +(p,s) = b†(p,s)b(p,s) ; N −(p,s) = d†(p,s)d(p,s)
and we can rewrite
P µ = dk kµ (N +(k,s) + N −(k,s)) s Z X f Q = dk (N +(k,s) N −(k,s)) − Z s ❐ X Using the anti-commutatorf relations it is now easy to verify that the theory is Lorentz invariant, that is (see Problems)
i[M µν , ψ]=(xµ∂ν xν∂µ)ψ +Σµνψ . − Jorge C. Rom˜ao TCA-2012 – 56 Microscopic causality
❐ The anti-commutation relations can be used to find the anti-commutation Lecture 1 relations at equal times for the fields. We get Canonical Quantization Scalar fields † 3 ψα(~x, t), ψ (~y, t) =δ (~x ~y)δαβ Lecture 2 { β } − Dirac field † † ψα(~x, t), ψβ(~y, t) = ψ (~x, t), ψ (~y, t) = 0 • Quantization { } { α β } • Causality • Feynman propagator ❐ These relations can be generalized to unequal times Electromagnetic field
Discrete Symmetries ψ (x), ψ† (y) = dp (p/ + m)γ0 e−ip·(x−y) ( p/ + m)γ0 eip·(x−y) { α β } αβ − − αβ Z h i = (i∂/ + m)γ0 i∆(x y) fx αβ − where the ∆(x y) function was defined before for the scalar field. − ❐ The fact that γ0 appears is due to the fact that in the above relation we took ψ† and not ψ. In fact, if we multiply on the right by γ0 we get
ψ (x), ψ (y) =(i∂/ + m) i∆(x y) { α β } x αβ − ψ (x), ψ (y) = ψ (x), ψ (y) = 0 { α β } { α β }
Jorge C. Rom˜ao TCA-2012 – 57 Microscopic causality
❐ We can easily verify the covariance of the above relations. We use Lecture 1 Canonical Quantization U(a, b)ψ(x)U −1(a, b) = S−1(a)ψ(ax + b) Scalar fields
Lecture 2 U(a, b)ψ(x)U −1(a, b) = ψ(ax + b)S(a) Dirac field • Quantization • Causality S−1γµS = aµ γν • Feynman propagator ν
Electromagnetic field Discrete Symmetries ❐ We get
U(a, b) ψ (x), ψ (y) U −1(a, b) = { α β } =S−1(a) ψ (ax + b), ψ (ay + b) S (a) ατ { τ λ } λβ =S−1(a)(i∂/ + m) i∆(ax ay)S (a) ατ ax τλ − λβ =(i∂/ + m) i∆(x y) αβ − where we have used the invariance of ∆(x y) and the result − −1 S i∂/axS = i∂/x
Jorge C. Rom˜ao TCA-2012 – 58 Microscopic causality
❐ For (x y)2 < 0 the anti-commutators vanish, because ∆(x y) also Lecture 1 vanishes.− This result allows us to show that any two observab−les built as Canonical Quantization
Scalar fields bilinear products of ψ e ψ commute for two spacetime points for which 2 Lecture 2 (x y) < 0. − Dirac field • Quantization ❐ We have • Causality • Feynman propagator Electromagnetic field ψα(x)ψβ(x), ψλ(y)ψτ (y) = Discrete Symmetries =ψ (x) ψ (x), ψ (y) ψ (y) ψ (x), ψ (y) ψ (x)ψ (y) α { β λ } τ − { α λ } β τ + ψ (y)ψ (x) ψ (x), ψ (y) ψ (y) ψ (y), ψ (x) ψ (x) λ α { β τ } − λ { τ α } β =0
for (x y)2 < 0. − ❐ In this way the microscopic causality is satisfied for the physical observables, such as the charge density or the momentum density.
Jorge C. Rom˜ao TCA-2012 – 59 Feynman propagator
❐ For the Dirac field, as in the case of the charged scalar field, there are two Lecture 1 ways of increasing the charge by one unit in x′ and decrease it by one unit in Canonical Quantization
Scalar fields x (note that the electron has negative charge). These ways are
Lecture 2 ′ ′ † Dirac field θ(t t) 0 ψβ(x )ψα(x) 0 • Quantization − | | • Causality ′ † ′ • θ(t t ) 0 ψ (x)ψ (x ) 0 Feynman propagator − | α β | Electromagnetic field
Discrete Symmetries In one case an electron of positive energy is created at ~x in the instant t, propagates until ~x′ where is annihilated at time t′ >t. In the other case a positron of positive energy is created in x′ and annihilated at x with t>t′. ❐ The Feynman propagator is obtained summing the two amplitudes. Due the exchange of ψβ and ψα there must be a minus sign between these two amplitudes. Multiplying by γ0, we get for the Feynman propagator,
S (x′ x) =θ(t′ t) 0 ψ (x′)ψ (x) 0 F − αβ − | α β | θ(t t′) 0 ψ (x)ψ (x′ ) 0 − − | β α | 0 T ψ (x′) ψ (x) 0 ≡ | α β | Jorge C. Rom˜ao TCA-2012 – 60 Feynman propagator
❐ We have defined the time ordered product for fermion fields, Lecture 1 Canonical Quantization T η(x)χ(y) θ(x0 y0)η(x)χ(y) θ(y0 x0)χ(y)η(x) . Scalar fields ≡ − − − Lecture 2 Dirac field ❐ Inserting in the definitio the expansions for ψ and ψ we get, • Quantization • Causality ′ ′ • Feynman propagator ′ ′ −ik·(x −x) ′ ik·(x −x) SF (x x)αβ = dk (k/ + m)αβθ(t t)e +( k/ + m)αβθ(t t )e Electromagnetic field − − − − Discrete Symmetries Z h i 4 fd k i(k/ + m) ′ = αβ e−ik·(x −x) (2π)4 k2 m2 + iε Z − 4 d k ′ = S (k) e−ik·(x −x) (2π)4 F αβ Z
where SF (k) is the Feynman propagator in momenta space. ❐ We can also verify that Feynman’s propagator is the Green function for the Dirac equation, that is (see Problems),
(i∂/ m) S (x′ x) = iδ δ4(x′ x) − λα F − αβ λβ − Jorge C. Rom˜ao TCA-2012 – 61 Canonical formalism for Electromagnetic Field
❐ The free electromagnetic field is described by the classical Lagrangian, Lecture 1 Canonical Quantization 1 = F F µν, where F = ∂ A ∂ A Scalar fields L −4 µν µν µ ν − ν µ Lecture 2 ❐ Dirac field The free field Maxwell equations are
Electromagnetic field αβ • Introduction ∂αF = 0 • Undefined metric • Feynman Propagator
Discrete Symmetries that correspond to the usual equations in 3-vector notation,
∂E~ ~ E~ =0 ; ~ B~ = ∇· ∇ × ∂t ❐ The other Maxwell equations are a consequence of the anti-symmetry of Fµν and can written as, 1 ∂ F αβ =0 ; F αβ = εαβµν F α 2 µν correspondinge to e ∂B~ ~ B~ =0 ; ~ E~ = ∇· ∇ × − ∂t
Jorge C. Rom˜ao TCA-2012 – 62 Canonical formalism for Electromagnetic Field
❐ Classically, the quantities with physical significance are the fields E~ e B~ , and Lecture 1 µ Canonical Quantization the potentials A are auxiliary quantities that are not unique due to the Scalar fields gauge invariance of the theory. Lecture 2 ❐ Dirac field In quantum theory the potentials Aµ are the ones playing the leading role as, Electromagnetic field for instance in the minimal prescription. We have therefore to formulate the • Introduction µ ~ ~ • Undefined metric quantum fields theory in terms of A and not of E and B. • Feynman Propagator µ Discrete Symmetries ❐ When we try to apply the canonical quantization to the potentials A we immediately run into difficulties. ❐ For instance, if we define the conjugate momentum as, ∂ πµ = L ˙ ∂(Aµ) we get
0 k ∂ ˙ k ∂A k 0 ∂ π = L = A k = E π = L = 0 ∂(A˙ k) − − ∂x ∂A˙ 0
Jorge C. Rom˜ao TCA-2012 – 63 Canonical formalism for Electromagnetic Field
❐ Therefore the conjugate momentum to the coordinate A0 vanishes and does Lecture 1 not allow us to use directly the canonical formalism. The problem has its Canonical Quantization origin in the fact that the photon, that we want to describe, has only two Scalar fields µ Lecture 2 degrees of freedom (positive or negative helicity) but we are using a field A µ Dirac field with four degrees of freedom. In fact, we have to impose constraints on A Electromagnetic field in such a way that it describes the photon. This problem can be addressed in • Introduction • Undefined metric three different ways: • Feynman Propagator
Discrete Symmetries ❐ 1) Radiation Gauge Historically, this was the first method to be used. It is based in the fact that it is always possible to choose a gauge, called the radiation gauge, where
A0 =0 ; ~ A~ = 0 ∇· that is, the potential A~ is transverse. These conditions in reduce the number of degrees of freedom to two, the transverse components of A~. It is then possible to apply the canonical formalism to these transverse components and quantize the electromagnetic field in this way. The problem with this method is that we loose explicit Lorentz covariance. It is then necessary to show that this is recovered in the final result. This method is followed in many text books, for instance in Bjorken and Drell. Jorge C. Rom˜ao TCA-2012 – 64 Canonical formalism for Electromagnetic Field
Lecture 1 ❐ 2) Quantization of systems with constraints Canonical Quantization
Scalar fields It can be shown that the electromagnetism is an example of an Hamilton Lecture 2 generalized system, that is a system where there are constraints among the Dirac field variables. The way to quantize these systems was developed by Dirac for Electromagnetic field systems of particles with n degrees of freedom. The generalization to • Introduction • Undefined metric quantum field theories is done using the formalism of path integrals. We will • Feynman Propagator study this method in Chapter 6, as it will be shown, this is the only method Discrete Symmetries that can be applied to non-abelian gauge theories, like the Standard Model. ❐ 3) Undefined metric formalism There is another method that works for the electromagnetism, called the formalism of the undefined metric, developed by Gupta and Bleuler. In this formalism, that we will study below, Lorentz covariance is kept, that is we will always work with the 4-vector Aµ, but the price to pay is the appearance of states with negative norm. We have then to define the Hilbert space of the physical states as a sub-space where the norm is positive. We see that in all cases, in order to maintain the explicit Lorentz covariance, we have to complicate the formalism. We will follow the book of Silvan Schweber.
Jorge C. Rom˜ao TCA-2012 – 65 Undefined metric formalism
❐ To solve the difficulty of the vanishing of π0, we will start by modifying the Lecture 1 Maxwell Lagrangian introducing a new term, Canonical Quantization
Scalar fields 1 µν 1 2 Lecture 2 = FµνF (∂ A) Dirac field L −4 − 2ξ ·
Electromagnetic field • Introduction where ξ is a dimensionless parameter. • Undefined metric • Feynman Propagator ❐ The equations of motion are now, Discrete Symmetries 1 Aµ 1 ∂µ(∂ A) = 0 ⊔⊓ − − ξ · and the conjugate momenta ∂ 1 πµ = L = F µ0 gµ0(∂ A) ∂A˙ µ − ξ · that is π0 = 1 (∂ A) − ξ · ( πk = Ek
Jorge C. Rom˜ao TCA-2012 – 66 Undefined metric formalism
❐ We remark that the above Lagrangian and the equations of motion, reduce Lecture 1 to Maxwell theory in the gauge ∂ A = 0. This why we say that our choice Canonical Quantization · Scalar fields corresponds to a class of Lorenz gauges with parameter ξ. With this abuse of language (in fact we are not setting ∂ A = 0, otherwise the problems Lecture 2 · Dirac field would come back) the value of ξ = 1 is known as the Feynman gauge and Electromagnetic field ξ = 0 as the Landau gauge. • Introduction • Undefined metric • Feynman Propagator ❐ From the equations of motion we get Discrete Symmetries (∂ A) = 0 ⊔⊓ · implying that (∂ A) is a massless scalar field. Although it would be possible to continue with· a general ξ, from now on we will take the case of the so-called Feynman gauge, where ξ = 1. Then the equation of motion coincide with the Maxwell theory in the Lorenz gauge. ❐ As we do not have anymore π0 = 0, we can impose the canonical commutation relations at equal times:
[πµ(~x, t),A (~y, t)] = igµ δ3(~x ~y) ν − ν − [Aµ(~x, t),Aν(~y, t)]=[πµ(~x, t),πν(~y, t)]=0
Jorge C. Rom˜ao TCA-2012 – 67 Undefined metric formalism
❐ Knowing that [Aµ(~x, t),Aµ(~y, t)]=0 at equal times, we can conclude that Lecture 1 the space derivatives of A also commute at equal times. Then, noticing Canonical Quantization µ
Scalar fields that
Lecture 2 πµ = A˙ µ + space derivatives Dirac field − Electromagnetic field • Introduction we can write instead of the previous commutation relations • Undefined metric • Feynman Propagator ˙ ˙ Discrete Symmetries [Aµ(~x, t),Aν(~y, t)]=[Aµ(~x, t), Aµ(~y, t)]=0
[A˙ (~x, t),A (~y, t)] = ig δ3(~x ~y) µ ν µν − ❐ If we compare these relations with the corresponding ones for the real scalar field, where the only one non-vanishing is,
[˙ϕ(~x, t),ϕ(~y, t)] = iδ3(~x ~y) − −
we see (gµν = diag(+, , , ) that the relations for space components are equal but they differ for− the− time− component. This sign will be the source of the difficulties previously mentioned.
Jorge C. Rom˜ao TCA-2012 – 68 Undefined metric formalism
❐ If, for the moment, we do not worry about this sign, we expand Aµ(x) Lecture 1 Canonical Quantization 3 Scalar fields Aµ(x) = dk a(k,λ)εµ(k,λ)e−ik·x + a†(k,λ)εµ∗(k,λ)eik·x Lecture 2 Z λ=0 Dirac field X Electromagnetic field µ f • Introduction where ε (k,λ) are a set of four independent 4-vectors that we assume to • Undefined metric real, without loss of generality. • Feynman Propagator Discrete Symmetries ❐ We will now make a choice for these 4-vectors. We choose εµ(1) and εµ(2) orthogonal to kµ and nµ, such that
µ ′ ′ ε (k,λ)ε (k,λ ) = δ ′ for λ,λ = 1, 2 µ − λλ After, we choose εµ(k, 3) in the plane (kµ,nµ) to nµ such that ⊥ εµ(k, 3)n =0 ; εµ(k, 3)ε (k, 3) = 1 µ µ − ❐ Finally we choose εµ(k, 0) = nµ. The vectors εµ(k, 1) and εµ(k, 2) are called transverse polarizations, while εµ(k, 3) and εµ(k, 0) longitudinal and scalar polarizations, respectively.
Jorge C. Rom˜ao TCA-2012 – 69 Undefined metric formalism
❐ In the frame where nµ = (1, 0, 0, 0) and ~k is along the z axis we have
Lecture 1 µ µ Canonical Quantization ε (k, 0) (1, 0, 0, 0) ; ε (k, 1) (0, 1, 0, 0) ≡ ≡ Scalar fields µ µ Lecture 2 ε (k, 2) (0, 0, 1, 0) ; ε (k, 3) (0, 0, 0, 1) ≡ ≡ Dirac field ❐ Electromagnetic field In general we can show that • Introduction ′ • Undefined metric ∗ ′ λλ λλ µ ∗ν µν • Feynman Propagator ε(k,λ) ε (k,λ ) = g , g ε (k,λ)ε (k,λ) = g · Discrete Symmetries λ ❐ Inserting the plane wave expansionX we get
′ [a(k,λ),a†(k′,λ′)] = gλλ 2k0(2π)3δ3(~k ~k′) − − showing, once more, that the quanta associated with λ = 0 has a commutation relation with the wrong sign. ❐ Before addressing this problem, we can verify we get for arbitrary times
[A (x),A (y)] = ig ∆(x, y) µ ν − µν showing the covariance of the theory. The function ∆(x y) is the same that was introduced before for scalar fields. − Jorge C. Rom˜ao TCA-2012 – 70 Undefined metric formalism
❐ Therefore, up to this point, everything is as if we had 4 scalar fields. There Lecture 1 is, however, the problem of the sign difference in one of the commutators. Canonical Quantization Scalar fields ❐ Let us now see what are the consequences of this sign. For that we Lecture 2 introduce the vacuum state defined by Dirac field
Electromagnetic field • Introduction a(k,λ) 0 = 0 λ = 0, 1, 2, 3 • Undefined metric | i • Feynman Propagator ❐ To see the problem with the sign we construct the one-particle state with Discrete Symmetries scalar polarization, that is
1 = dk f(k)a†(k, 0) 0 | i | i Z and calculate itsf norm
1 1 = dk dk f ∗(k )f(k ) 0 a(k , 0)a†(k , 0) 0 h | i 1 2 1 2 | 1 2 | Z
= f0 0f dk f(k) 2 −h | i | | Z The state 1 has a negativef norm. | i Jorge C. Rom˜ao TCA-2012 – 71 Undefined metric formalism
❐ The same calculation for the other polarization would give well behaved Lecture 1 positive norms. We therefore conclude that the Fock space of the theory has Canonical Quantization
Scalar fields indefinite metric. What happens to the probabilistic interpretation of QM? Lecture 2 ❐ To solve this problem we note that we are not working anymore with the Dirac field
Electromagnetic field classical Maxwell theory because we modified the Lagrangian. What we • Introduction would like to do is to impose the condition ∂ A = 0, but that is impossible • Undefined metric · • Feynman Propagator as an equation for operators. We can, however, require that condition on a Discrete Symmetries weaker form, as a condition only to be verified by the physical states.
❐ More specifically, we require that the part of ∂ A that contains the annihilation operator (positive frequencies) annihilate· s the physical states,
∂µA(+) ψ = 0 µ | i The states ψ can be written in the form | i ψ = ψ φ | i | T i| i
where ψT is obtained from the vacuum with creation operators with transverse| i polarization and φ with scalar and longitudinal polarization. | i Jorge C. Rom˜ao TCA-2012 – 72 Undefined metric formalism
❐ This decomposition depends, of course, on the choice of polarization vectors. Lecture 1 To understand the consequences it is enough to analyze the states φ as Canonical Quantization µ (+) | i Scalar fields ∂ Aµ contains only scalar and longitudinal polarizations,
Lecture 2 Dirac field i∂ A(+) = dk e−ik·x a(k,λ) ε(k,λ) k Electromagnetic field · · • Introduction Z λ=0,3 • Undefined metric X • Feynman Propagator f ❐ Discrete Symmetries Therefore the previous condition becomes
k ε(k,λ) a(k,λ) φ = 0 · | i λX=0,3 ❐ This does not determine completely φ . In fact, there is much arbitrariness in the choice of the transverse polarization| i vectors, to which we can always add a term proportional to kµ because k k = 0. This arbitrariness must reflect itself on the choice of φ . The condition· is equivalent to, | i [a(k, 0) a(k, 3)] φ = 0 . − | i
Jorge C. Rom˜ao TCA-2012 – 73 Undefined metric formalism
❐ We can construct φ as a linear combination of states φn with n scalar or Lecture 1 longitudinal photons:| i | i Canonical Quantization
Scalar fields φ = C φ + C φ + + C φ + Lecture 2 | i 0 | 0i 1 | 1i ··· n | ni ··· Dirac field φ0 0 Electromagnetic field | i≡| i • Introduction • Undefined metric • Feynman Propagator ❐ The states φ are eigenstates of the operator number for scalar or | ni Discrete Symmetries longitudinal photons,
N ′ φ = n φ | ni | ni where
N ′ = dk a†(k, 3)a(k, 3) a†(k, 0)a(k, 0) − Z f ❐ Then
n φ φ = φ N ′ φ = 0 h n| ni h n| | ni
Jorge C. Rom˜ao TCA-2012 – 74 Undefined metric formalism
❐ This means that Lecture 1
Canonical Quantization φn φn = δn0 Scalar fields h | i Lecture 2 that is, for n = 0, the state φn has zero norm. We have then for the Dirac field 6 | i Electromagnetic field general state φ , • Introduction | i • Undefined metric φ φ = C 2 0 • Feynman Propagator h | i | 0| ≥ Discrete Symmetries and the coefficients C , i = 1, n are arbitrary. i ··· ··· ❐ We have to show that this arbitrariness does not affect the physical observables. The Hamiltonian is
H = d3x : πµA˙ : µ −L Z 3 1 = d3x : A˙ 2 +(~ A )2 A˙ 2 (~ A )2 : 2 i ∇ i − 0 − ∇ 0 i=1 Z X h i 3 = dk k0 a†(k,λ)a(k,λ) a†(k, 0)a(k, 0) " − # Z λX=1 Jorge C. Rom˜ao f TCA-2012 – 75 Undefined metric formalism
❐ It is easy to check that if ψ is a physical state we have Lecture 1 | i Canonical Quantization ψ dk k0 2 a†(k,λ)a(k,λ) ψ Scalar fields ψ H ψ T | λ=1 | T Lecture 2 h | | i = ψ ψ D R P ψT ψT E Dirac field h | i f h | i Electromagnetic field • Introduction and the arbitrariness on the physical states completely disappears when we • Undefined metric take average values. Besides that, only the physical transverse polarizations • Feynman Propagator contribute to the result. One can show that the arbitrariness in φ is related Discrete Symmetries with a gauge transformation within the class of Lorenz gauges. | i ❐ It is important to note that although for the average values of the physical observables only the transverse polarizations contribute, the scalar and longitudinal polarizations are necessary for the consistency of the theory. In particular they show up when we consider complete sums over the intermediate states. ❐ Invariance for translations is readily verified. For that we write,
3 P µ = dk kµ ( gλλ)a†(k,λ)a(k,λ) − Z λX=0 Jorge C. Rom˜ao f TCA-2012 – 76 Undefined metric formalism
❐ Then Lecture 1
Canonical Quantization ′ µ ν µ λλ † ′ ′ ν ′ ′ −ik·x Scalar fields i[P ,A ] = dk dk ik ( g ) a (k,λ)a(k,λ),a(k ,λ ) ε (k ,λ )e ′ − Lecture 2 Z λ,λ X Dirac field f f ′ + a†(x,λ)a(k,λ),a†(k′,λ′) ε∗ν (k′,λ′)eik ·x Electromagnetic field • Introduction o • Undefined metric µ ν −ik·x † ν ik·x • = dk ik a(k,λ)ε (k,λ)e a (k,λ)ε (k,λ)e Feynman Propagator − Discrete Symmetries Z λ X f =∂µAν
showing the invariance under translations. ❐ In a similar way, it can be shown the invariance for Lorentz transformations (see Problems). For that we have to show that
M jk = d3x : xjT 0k xkT 0j + EjAk EkAj : − − Z M 0i = d3x : x0T 0i xiT 00 (∂ A)Ai EiA0 : − − · − Z Jorge C. Rom˜ao TCA-2012 – 77 Undefined metric formalism
❐ Where (ξ = 1) Lecture 1 Canonical Quantization T 0i = (∂ A) ∂iA0 Ek∂iAk Scalar fields − · − Lecture 2 3 Dirac field 00 ˙ 2 ~ 2 ˙ 2 ~ 2 T = Ai +( Aj) A0 ( A0) Electromagnetic field ∇ − − ∇ • Introduction i=1 • Undefined metric X h i • Feynman Propagator ❐ Discrete Symmetries Using these expressions one can show that the photon has helicity 1, corresponding therefore to spin one. For that we start by choosing± the direction of ~k along the axis 3 (z axis) and take the polarization vector defined before. A one-photon physical state will then be (not normalized),
k,λ = a†(k,λ) 0 λ = 1, 2 | i | i ❐ Let us now calculate the angular momentum along the axis 3. This is given by
M 12 k,λ = M 12a†(k,λ) 0 =[M 12,a†(k,λ)] 0 | i | i | i where we have used the fact that the vacuum state satisfies M 12 0 = 0. | i Jorge C. Rom˜ao TCA-2012 – 78 Undefined metric formalism
❐ The operator M 12 has one part corresponding the orbital angular momenta Lecture 1 and another corresponding to the spin. The contribution of the orbital Canonical Quantization
Scalar fields angular momenta vanishes (angular momenta in the direction of motion) as Lecture 2 one can see calculating the commutator. In fact the commutator with the 1 2 Dirac field orbital angular momenta is proportional to k or k , which are zero by Electromagnetic field hypothesis. Let us then calculate the spin part. Using the notation, • Introduction • Undefined metric − • Feynman Propagator Aµ = Aµ(+) + Aµ( ) Discrete Symmetries where Aµ(+)(Aµ(−)) correspond to the + ( ) frequencies, we get − : E1A2 E2A1 := E1(+)A2(+)+E1(−)A2(+)+A2(−)E1(+)+E1(−)A2(−) (1 2) − − ↔ ❐ Then
: E1A2 E2A1 :,a†(k,λ) = − =E1(+) A2(+),a†(k,λ ) + E1(+),a†(k,λ) A2(+) h i + E1( ) A2(+),a†(k,λ ) + A2(−) E1(+),a†(k,λ) (1 2) − − ↔ h i =E1 A2(+),a†(k,λ) + A2 E1(+),a†(k,λ) (1 2) − ↔ Jorge C. Rom˜ao h i h i TCA-2012 – 79 Undefined metric formalism
❐ Now (recall that λ = 1, 2) Lecture 1
Canonical Quantization ′ ′ 2(+) † 2 ′ ′ ′ ′ † −ik ·x Scalar fields [A ,a (k,λ)] = dk ε (k ,λ ) a(k ,λ ),a (k,λ) e Lecture 2 Z λ′ X Dirac field =ε2(k,λf )e−ik·x Electromagnetic field • Introduction ′ ′ • Undefined metric 1(+) † ′0 0 ′ ′ ′1 0 ′ ′ ′ ′ † −ik ·x • Feynman Propagator [E ,a (k,λ)] = dk ik ε (k ,λ )+ik ε (k ,λ ) a(k ,λ ),a (k,λ) e ′ Discrete Symmetries Z λ X =ik0fε1(k,λ)e−ik·x ❐ Therefore
d3x : E1A2 E2A1 :,a†(k,λ) − Z = d3xe−ik·x E1ε2(k,λ) + A2ik0ε1(k,λ) E2ε1(k,λ) + A1ik0ε2(k,λ) − Z ↔ ↔ = d3xe−ik·x ε1(k,λ)∂ A2(x) ε2(k,λ)∂ A1(x) 0 − 0 Z h i where we have used the fact that Ei = A˙ i, i = 1, 2, for our choice of frame and polarization vectors. − Jorge C. Rom˜ao TCA-2012 – 80 Undefined metric formalism
❐ On the other hand Lecture 1
Canonical Quantization ↔ a(k,λ) = i d3xeik·x ∂ εµ(k,λ)A (x) Scalar fields − 0 µ Lecture 2 Z Dirac field ↔ † 3 −ik·x µ Electromagnetic field a (k,λ) = i d xe ∂ 0 ε (k,λ)Aµ(x) • Introduction • Undefined metric Z • Feynman Propagator ❐ For our choice we get Discrete Symmetries ↔ a†(k, 1) = i d3xe−ik·x ∂ A1(x) − 0 Z ↔ a†(k, 2) = i d3xe−ik·x ∂ A2(x) − 0 Z ❐ Therefore
[M 12,a†(k,λ)] = iε1(k,λ)a†(k, 2) iε2(k,λ)a†(k, 1) −
Jorge C. Rom˜ao TCA-2012 – 81 Undefined metric formalism
❐ We find that the state a†(k,λ) 0 ,λ = 1, 2 is not an eigenstate of the Lecture 1 12 | i Canonical Quantization operator M . Scalar fields ❐ However the linear combinations, Lecture 2
Dirac field † 1 † † Electromagnetic field aR(k) = a (k, 1) + ia (k, 2) • Introduction √2 • Undefined metric • Feynman Propagator † 1 † † Discrete Symmetries a (k) = a (k, 1) ia (k, 2) L √2 − which correspond to right and left circular polarization, verify
[M 12,a† (k)] = a† (k); [M 12,a† (k)] = a† (k) R R L − L ❐ This shows that the photon has spin 1 with right or left circular polarization (negative or positive helicity).
Jorge C. Rom˜ao TCA-2012 – 82 Feynman propagator
Lecture 1 ❐ The Feynman propagator is defined as the vacuum expectation value of the Canonical Quantization time ordered product of the fields, that is Scalar fields
Lecture 2 Gµν(x, y) 0 TAµ(x)Aν(y) 0 Dirac field ≡h | | i 0 0 0 0 Electromagnetic field =θ(x y ) 0 Aµ(x)Aν(y) 0 + θ(y x ) 0 Aν (y)Aµ(x) 0 • Introduction − h | | i − h | | i • Undefined metric • Feynman Propagator ❐ Inserting the expansions for Aµ(x) and Aν(y) we get Discrete Symmetries
0 0 G (x y) = g dk e−ik·(x−y)θ(x0 y0) + eik·(x−y)θ(y −x ) µν − − µν − Z d4kh i i = g f e−ik·(x−y) − µν (2π)4 k2 + iε Z d4k G (k)e−ik·(x−y) ≡ (2π)4 µν Z ❐ Gµν(k) is the Feynman propagator on the momentum space ig G (k) − µν µν ≡ k2 + iε Jorge C. Rom˜ao TCA-2012 – 83 Feynman propagator
❐ It is easy to verify that Gµν(x y) is the Green’s function of the equation of Lecture 1 motion, that for ξ = 1 is the wave− equation, that is Canonical Quantization
Scalar fields G (x y) = ig δ4(x y) Lecture 2 ⊔⊓x µν − µν − Dirac field ❐ These expressions for Gµν(x y) and Gµν(k) correspond to the particular Electromagnetic field − • Introduction case of ξ = 1, the so-called Feynman gauge. For the general case, ξ = 0 • Undefined metric 6 • Feynman Propagator µ 1 µ ρ Discrete Symmetries g 1 ∂ ∂ A (x) = 0 ⊔⊓x ρ − − ξ ρ ❐ For this case the equal times commutation relations are more complicated (see Problems). Using those relations one can show that the Feynman propagator is still the Green’s function of the equation of motion, that is
1 gµ 1 ∂µ∂ 0 TAρ(x)Aν(y) 0 = igµν δ4(x y) ⊔⊓x ρ − − ξ ρ h | | i − ❐ Using this equation we can then obtain in an arbitrary ξ gauge (of the Lorenz type), g k k G (k) = i µν + i(1 ξ) µ ν . µν − k2 + iε − (k2 + iε)2
Jorge C. Rom˜ao TCA-2012 – 84 Discrete Symmetries
❐ We know from the study of the Dirac equation the transformations like Lecture 1 space inversion (Parity) and charge conjugation, are symmetries of the Dirac Canonical Quantization
Scalar fields equation. Lecture 2 ❐ More precisely, if ψ(x) is a solution of the Dirac equation, then Dirac field Electromagnetic field ′ ′ Discrete Symmetries ψ (x) = ψ ( ~x, t) = γ0ψ(~x, t) • Parity − • Charge conjugation c T • Time reversal ψ (x) = Cψ (x) • TCP are also solutions (if we take the charge e for ψc). Similar operations could also be defined for scalar and vector fields.− ❐ With second quantization the fields are no longer functions, they become operators. We have therefore to find unitary operators and that describe those operations within this formalism. There is anotherP discreteC symmetry, time reversal, that in second quantization will be described by an anti-unitary operator . T ❐ We will exemplify with the scalar field how to get these operators. We will leave the Dirac and Maxwell fields as exercises.
Jorge C. Rom˜ao TCA-2012 – 85 Parity
❐ To define the meaning of the Parity operation we have to put the system in Lecture 1 interaction with the measuring system, considered to be classical. This Canonical Quantization
Scalar fields means that we will consider the system described by
Lecture 2 j (x)Aµ (x) Dirac field L −→ L − µ ext Electromagnetic field Discrete Symmetries where we have considered that the interaction is electromagnetic. jµ(x) is • Parity the electromagnetic current that has the form, • Charge conjugation • Time reversal ↔ • TCP ∗ jµ(x) = ie : ϕ ∂ µϕ : scalar field
jµ(x) = e : ψγµψ : Dirac field ❐ In a Parity transformation we invert the coordinates of the measuring system, therefore the classical fields are now
Aµ =(A0 ( ~x, t)), A~ ( ~x, t) = Aext( ~x, t) ext ext − − ext − µ − ❐ For the dynamics of the new system to be identical to that of the original system, which should be the case if Parity is conserved, it is necessary that the equations of motion remain the same.
Jorge C. Rom˜ao TCA-2012 – 86 Parity
❐ This is true if Lecture 1 Canonical Quantization (~x, t) −1 = ( ~x, t) Scalar fields PL P L − Lecture 2 −1 µ jµ(~x, t) = j ( ~x, t) Dirac field P P − Electromagnetic field Discrete Symmetries ❐ These are the conditions that a theory should obey in order to be invariant • Parity • Charge conjugation under Parity. • Time reversal • TCP ❐ Furthermore should leave the commutation relations unchanged, so that the quantumP dynamics is preserved. For each theory that conserves Parity should be possible to find an unitary operator that satisfies these conditions. P ❐ Now we will find such an operator for the scalar field. It is easy to verify that the condition P
ϕ(~x, t) −1 = ϕ( ~x, t) P P ± − satisfies all the requirements. The sign is the intrinsic parity of the particle described by the field ϕ,(+ for± scalar and for pseudo-scalar). − Jorge C. Rom˜ao TCA-2012 – 87 Parity
❐ In terms of the expansion of the momentum, the requirement is Lecture 1 Canonical Quantization a(k) −1 = a( k) ; a†(k) −1 = a†( k) Scalar fields P P ± − P P ± − Lecture 2 where k means that we have changed ~k into ~k (but k0 remains intact, Dirac field − − Electromagnetic field that is, k0 = + ~k 2 + m2). Discrete Symmetries | | • Parity ❐ q • Charge conjugation It is easier to solve this requirement in the momentum space. As should • P Time reversal be unitary, we write • TCP = eiP P ❐ Then in a(k) −1 =a(k) + i[P,a(k)] + + [P, [ , [P,a(k)] ] + P P ··· n! ··· ··· ··· = a( k) − − where we have chosen the case of the pseudo-scalar field.
Jorge C. Rom˜ao TCA-2012 – 88 Parity
❐ The last relation suggests the form Lecture 1 Canonical Quantization λ Scalar fields [P,a(k)] = [a(k) + εa( k)] 2 − Lecture 2 Dirac field where λ and ε = 1 are to be determined. Electromagnetic field ± Discrete Symmetries ❐ We get • Parity • Charge conjugation 2 • Time reversal λ • TCP [P, [P,a(k)]] = [a(k) + εa( k)] 2 − and therefore 1 (iλ)2 (iλ)4 a(k) −1 =a(k) + iλ + + + + (a(k) + εa( k)) P P 2 2! ··· n! ··· − 1 1 = [a(k) εa( k)] + eiλ[a(k) + εa( k)] 2 − − 2 − = a( k) − − ❐ We solve this if we choose λ = π and ε =+1 (λ = π and ε = 1 for the scalar case). −
Jorge C. Rom˜ao TCA-2012 – 89 Parity
❐ It is easy to check that (for λ = π and ε = +1)
Lecture 1 Canonical Quantization π † † † Pps = dk a (k)a(k) + a (k)a( k) = Pps Scalar fields − 2 − Lecture 2 Z π † † Dirac field =exp if dk a (k)a(k) + a (k)a( k) Pps − 2 − Electromagnetic field Z Discrete Symmetries ❐ For the scalar field • Parity f • Charge conjugation π • Time reversal † † s = exp i dk a (k)a(k) a (k)a( k) • TCP P − 2 − − Z ❐ For the case of the Dirac field, the condition is now f ψ(~x, t) −1 = γ0ψ( ~x, t) P P − ❐ Repeating the same steps we get
π =exp i dp b†(p,s)b(p,s) b†(p,s)b( p,s) PDirac − 2 − − ( Z s X + d†(p,sf )d(p,s) + d†(p,s)d( p,s) − ❐ The case of the Maxwell field is left as an exercise. io Jorge C. Rom˜ao TCA-2012 – 90 Charge conjugation
❐ The conditions for charge conjugation invariance are now Lecture 1 Canonical Quantization −1 −1 (x) = ; jµ = jµ Scalar fields CL C L C C − Lecture 2 where j is the electromagnetic current. Dirac field µ Electromagnetic field ❐ These conditions are verified for the charged scalar fields if Discrete Symmetries • Parity −1 ∗ ∗ −1 • Charge conjugation ϕ(x) = ϕ (x) ; ϕ (x) = ϕ(x) • Time reversal C C C C • TCP and for the Dirac field if
ψ (x) −1 = C ψ (x) C α C αβ β ψ (x) −1 = ψ (x)C−1 C α C − β βα where C is the charge conjugation matrix.
µ ❐ Finally from the invariance of jµA we obtain the condition for the electromagnetic field,
A −1 = A C µC − µ Jorge C. Rom˜ao TCA-2012 – 91 Charge conjugation
❐ By using a method similar to the one used in the case of the Parity we can Lecture 1 get the operator for the different theories. For the scalar field we get Canonical Quantization C Scalar fields π † † Lecture 2 = exp i dk (a a )(a a−) Cs 2 + − − + − Dirac field Z Electromagnetic field ❐ For the Dirac field f Discrete Symmetries • Parity • Charge conjugation = 1 2 • Time reversal C C C • TCP with
=exp i dp φ(p,s) b†(p,s)b(p,s) d†(p,s)d(p,s) C1 − − ( Z s ) X π f =exp i dp b†(p,s) d†(p,s) [b(p,s) d(p,s)] C2 2 − − ( Z s ) ❐ X Where f v(p,s) = eiφ(p,s) uc(p,s) , u(p,s) = eiφ(p,s) vc(p,s)
and the phase φ(p,s) is arbitrary.
Jorge C. Rom˜ao TCA-2012 – 92 Time reversal
❐ Classically the meaning of the time reversal invariance it is clear. We change Lecture 1 the sign of the time, the velocities change direction and the system goes Canonical Quantization
Scalar fields from what was the final state to the initial state. Lecture 2 ❐ This exchange between the initial and final state has as consequence, in Dirac field quantum mechanics, that the corresponding operator must be anti-linear or Electromagnetic field ∗ Discrete Symmetries anti-unitary. In fact f i = i f and therefore if we want • Parity h | i h | i ϕf ϕi = ϕi ϕf then must include the complex conjugation • Charge conjugation hT |T i h | i T • Time reversal operation. • TCP ❐ We can write
= K T U where is unitary and K is the instruction to tale the complex conjugate of all c-numbersU . Then Tϕ Tϕ = Kϕ Kϕ h f | ii hU f |U ii = ϕ ϕ ∗ hU f |U ii = ϕ ϕ ∗ = ϕ ϕ h f | ii h i| f i as we wanted. Jorge C. Rom˜ao TCA-2012 – 93 Time reversal
❐ A theory will be invariant under time reversal if Lecture 1 Canonical Quantization −1 −1 µ (~x, t) = (~x, t) , jµ(~x, t) = j (~x, t) Scalar fields TL T L − T T − Lecture 2 ❐ Dirac field For the scalar field this condition will be verified if
Electromagnetic field −1 Discrete Symmetries ϕ(~x, t) = ϕ(~x, t) • Parity T T ± − • Charge conjugation • Time reversal ❐ For the electromagnetic field we must have. • TCP Aµ(~x, t) −1 = A (~x, t) T T µ − µ making j Aµ invariant. ❐ For the case of the Dirac field the transformation is
ψ (~x, t) −1 = T ψ (~x, t) T α T αβ β − In order that the last equation is satisfied, the T matrix must satisfy
−1 T µ∗ TγµT = γµ = γ
Jorge C. Rom˜ao TCA-2012 – 94 Time reversal
❐ This has a solution, in the Dirac representation, Lecture 1 Canonical Quantization T = iγ1γ3 Scalar fields
Lecture 2 Dirac field ❐ Applying the same type of reasoning already used for and we can find P C Electromagnetic field , or equivalently, . For the Dirac field, noticing that Discrete Symmetries T U • Parity ∗ iα+(p,s) • Charge conjugation Tu(p,s) =u ( p, s)e • Time reversal − − • TCP T v(p,s) =v∗( p, s)eiα−(p,s) − − we can write = and obtain U U1U2
† † =exp i dp α b (p,s)b(p,s) α−d (p,s)d(p,s) U1 − + − ( Z s ) X π f =exp i dp b†(p,s)b(p,s) + b†(p,s)b( p s) U2 − 2 − − ( Z s X f d†(p,s)d(p,s) d†(p,s)d( p, s) − − − − #) Jorge C. Rom˜ao TCA-2012 – 95 The theorem TCP
Lecture 1 ❐ It is a fundamental theorem in Quantum Field Theory that the product Canonical Quantization is an invariance of any theory that satisfies the following general conditions:T CP Scalar fields
Lecture 2 ◆ The theory is local and covariant for Lorentz transformations. Dirac field Electromagnetic field ◆ The theory is quantized using the usual relation between spin and Discrete Symmetries statistics, that is, commutators for bosons and anti-commutators for • Parity • Charge conjugation fermions. • Time reversal • TCP ❐ This theorem due to L¨udus, Zumino, Pauli e Schwinger has an important consequence that if one of the discrete symmetries is not preserved then another one must also be violated to preserve the invariance of the product. ❐ For a proof of the theorem see, for instance, the books of Bjorken and Drell and Itzykson and Zuber
Jorge C. Rom˜ao TCA-2012 – 96