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Advanced Quantum Field Theory Chapter 1 Canonical Quantization of Free Fields

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Advanced Quantum Field Theory Chapter 1 Canonical Quantization of Free Fields 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) .
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