MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof. Alan Guth February 16, 2008 LECTURE NOTES 1 QUANTIZATION OF THE FREE SCALAR FIELD As we have already seen, a free scalar field can be described by the Lagrangian L = d3 x � , (1) where 1 1 � =2 ∂µφ∂µφ − 2 m 2φ2 (2a) =1 φ˙2 − 1 ∇iφ ∇iφ − 1 m 2φ2 . (2b) 2 2 2Our goal is to “quantize” this theory, in the sense of developing a quantum theory that corresponds to the classical theory described by the above Lagrangian. 1. CANONICAL QUANTIZATION: Here we will use the method of canonical quantization, which I assume is already familiar to you in the context of quantum mechanics. Specifically, suppose that we were given a Lagrangian with a discrete number of dynamical variables qi: L = L(qi, q˙i,t) . (3) The canonical momenta would then be defined by ∂L pi ≡ , (4)∂q˙i and the Hamiltonian would be given by H = pi q˙i − L. (5) i A quantum theory corresponding to this classical theory could then be constructed by promoting each qi and pi to an operator on a Hilbert space, and insisting on the canonical commutation relations [qi ,pj ]= i¯hδij . (6)� � � � � � � 8.323 LECTURE NOTES 1, SPRING 2008: Quantization of the Free Scalar Field p. 2 For most of this course we will use units for which ¯h ≡ 1, but for now I will leave the ¯h’s in the equations. The Hamiltonian H(pi,qi) is then also an operator on the Hilbert space, and in the Schr¨odinger picture the physical states evolve according to the Schr¨odinger equation ∂ i¯h |ψ(t) = H |ψ(t) . (7)∂t If H is independent of time, Eq. (7) has the formal solution |ψ(t) = e −iHt/h¯|ψ(0) . (8) Given any operator � , its expectation value in the state |ψ(t) is then given by ψ(t) |� | ψ(t) = ψ(0) � eiHt/h¯� e −iHt/h¯� ψ(0) . (9) This equation leads naturally to the Heisenberg picture description, in which the states are treated as time-independent, and all of the time dependence is incorporated into the evolution of the operators: � (t)= eiHt/h¯� e −iHt/h¯. (10) 2. FIELD QUANTIZATION BY LATTICE APPROXIMATION: To quantize the classical field theory of Eq. (2), we can begin by quantizing a lattice version of the theory. That is, we can replace the continuous space by a cubic lattice of closely spaced grid points, with a lattice spacing a, and we can truncate the space to a finite region. The system then reduces to one with a discrete number of dynamical variables, exactly like the systems that we already know how to quantize. Then if we can take the limit as the lattice spacing a approaches zero and the volume approaches infinity, the quantization of the field theory can be completed. We will see later that the a → 0 limit is problematic for interacting theories, but we will see here that this program can be carried out easily for the free theory. When we replace the continuous space by a finite lattice of points, we can label each lattice site with an index k. In a fully detailed lattice description we would probably label each lattice site with a triplet of integers representing the x, y,and z coordinates of the site, but for present purposes it will suffice to imagine simply numbering all the lattice sites from 1 to N,where N is the total number of sites. The field φ(x, t) is then replaced by a set of dynamical variables φk(t), where one can think of φk (t)as representing the average value of φ(x, t) in a cube of size a surrounding the lattice site k. The Lagrangian of Eqs. (1) and (2) is then replaced by L = � k ∆V, (11) k� � � � 8.323 LECTURE NOTES 1, SPRING 2008: Quantization of the Free Scalar Field p. 3 where ∆V = a 3 (12) and � k = 1 2 ˙φ2 k − 1 2 ∇iφk ∇iφk − 1 2 m 2φ2 k . (13) Here the lattice derivative ∇iφk is defined by ∇iφk ≡ φk� (k,i) − φk , (14) a where k(k, i) denotes the lattice site that is a distance a in the ith direction from the lattice site k. The canonical momenta are then given by pk = ∂L = ∂� k ∆V = φ˙k ∆V. (15) ∂φ˙k ∂φ˙k Since the canonical momenta are proportional to ∆V , it is natural to define a canonical momentum density πk by πk ≡ ∆pk V = ∂∂� φ˙kk = φ˙k . (16) Following Eq. (5), the Hamiltonian is then H = pkφ˙k − L = πkφ˙k − � k ∆V. (17) k k The canonical commutation relations become [φk� ,φk]=0 , [pk� ,pk]=0 , and [φk� ,pk]= ihδ¯k�k . (18) In terms of the canonical momentum densities, i¯k�khδ[φk� ,φk]=0 , [πk� ,πk]=0 , and [φk� ,πk]= . (19)∆V Although we have not yet constructed the full theory, it is not too early to write down the continuum limit of these defining equations. The continuum canonical momentum density becomes π(x, t)= ∂� = φ˙(x, t) , (20) ∂φ˙(x, t)� � � � � � � � � � 8.323 LECTURE NOTES 1, SPRING 2008: Quantization of the Free Scalar Field p. 4 and the Hamiltonian becomes H = d3 x πφ˙− � . (21) The trivial canonical commutation relations carry over trivially: [φ(x ,t) ,φ(x, t)] = 0 and [π(x ,t) ,π(x, t)] = 0 , (22) obviously. For the nontrivial commutation relation, the result will be clearest if we first rewrite the last equation in (19) as a sum which will become an integral in the limit. If we let � denote a region of the lattice, the last equation in (19) becomes � � 1if k ∈ �[φk� ,πk]∆V = i¯h δk� ,k = ih¯ (23)0otherwise. k∈ � k∈ � In the continuum limit [φk� ,πk]∆V k∈ � clearly approaches d3 x [φ(x ,t) ,π(x, t)] , x∈ � so Eq. (23) becomes d3 x [φ(x ,t) ,π(x, t)] = ih¯1if x ∈ � (24) x∈ � 0otherwise. This relationship can be expressed more conveniently by introducing the Dirac delta-function δ3(x), which is defined by its integral*: d3xf(x) δ(x − x ) ≡ f(x)if x ∈ � (25) x∈ � 0 otherwise. Given this definition, Eq. (24) can be rewritten as [φ(x x, t)] = i¯ x − ) .,t) ,π( hδ( x (26) Note that the delta function is