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Lecture I: Collective Excitations: from Particles to Fields Free Scalar Field Theory: Phonons

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Lecture I: Collective Excitations: from Particles to Fields Free Scalar Field Theory: Phonons Lecture I 1 Lecture I: Collective Excitations: From Particles to Fields Free Scalar Field Theory: Phonons The aim of this course is to develop the machinery to explore the properties of quantum systems with very large or infinite numbers of degrees of freedom. To represent such systems it is convenient to abandon the language of individual elementary particles and speak about quantum fields. In this lecture, we will consider the simplest physical example of a free or non-interacting many-particle theory which will exemplify the language of classical and quantum fields. Our starting point is a toy model of a mechanical system describing a classical chain of atoms coupled by springs. Discrete elastic chain RI-1 φ I M ks (I-1)a Ia (I+1)a Equilibrium positionx ¯n na; natural length a; spring constant ks ≡ Goal: to construct and quantise a classical field theory for the collective (longitundinal) vibrational modes of the chain . Discrete Classical Lagrangian: k:e: p:e: in spring N z }| { z }| { X m 2 ks 2 L = T V = x_ (xn+1 xn a) − 2 n − 2 − − n=1 assume periodic boundary conditions (p.b.c.) xN+1 = Na + x1 (and setx _ n @txn) ≡ Using displacement from equilibrium φn = xn x¯n − N m k X _2 s 2 L = φ (φn+1 φn) ; p:b:c : φN+1 φ1 2 n − 2 − ≡ n=1 In principle, one can obtain exact solution of discrete equation of motion | see PS I However, typically, one is not concerned with behaviour on `atomic' scales: 1. for such purposes, modelling is too primitive! viz. anharmonic contributions 2. such properties are in any case `non-universal' Aim here is to describe low-energy collective behaviour | generic, i.e. universal In this case, it is often permissible to neglect the discreteness of the microscopic entities of the system and to describe it in terms of effective continuum degrees of freedom. Lecture Notes October 2006 Lecture I 2 . Continuum Lagrangian Describe φn as a smooth function φ(x) of a continuous variable x; makes sense if φn+1 φn a (i.e. gradients small) − Z L=Na 1=2 3=2 X 1 φn a φ(x) ; φn+1 φn a @xφ(x) ; dx a ! x=na − ! x=na n −! 0 N.B. [φ(x)] = L1=2 φ(x) φI RI Lagrangian functional Lagrangian density z }| { Z L z }| 2 { m 2 ksa 2 L[φ] = dx (φ, @xφ, φ_); (φ, @xφ, φ_) = φ_ (@xφ) 0 L L 2 − 2 . Classical action Z Z Z L S[φ] = dt L[φ] = dt dx (φ, @xφ, φ_) 0 L N-point particle degrees of freedom continuous classical field φ(x) • 7! Dynamics of φ(x) specified by functionals L[φ] and S[φ] • What are the corresponding equations of motion...? ||||||||||||||{ . Hamilton's Extremal Principle: (Revision) Z Suppose classical point particle x(t) described by action S[x] = dt L(x; x_) Configurations x(t) that are realised are those that extremise the action i.e. for any smooth function η(t), the \variation", 1 δS[x] lim!0 (S[x + η] S[x]) = 0 is stationary ≡ − ; Euler-Lagrange equations of motion Z t Z t 2 S[x + η] = dt L(x + η; x_ + η_) = dt (L(x; x_) + η@xL + η@_ x_ L) + O( ) 0 0 = 0 z }| { Z by parts Z d δS[x] = dt (η@xL +η@ _ x_ L) = dt @xL (@x_ L) η = 0 − dt t Note: boundary term, η@x_ L vanishes by construction j0 Lecture Notes October 2006 Lecture I 3 "#(x,t) ! !(x,t) x t . Generalisation to continuum field x φ(x)? 7! Apply same extremal principle: φ(x; t) φ(x; t) + η(x; t) with7! both φ and η periodic in x, i.e. φ(x + L) = φ(x) Z t Z L 2 2 S[φ + η] = S[φ] + dt dx mφ_η_ ksa @xφ∂xη + O( ): 0 0 − t L Integrating by parts boundary terms vanish by construction: ηφ_ = 0 = η@xφ j0 j0 Z t Z L ¨ 2 2 δS = dt dx mφ ksa @xφ η = 0 − 0 0 − 2 2 2 Since η(x; t) is an arbitrary smooth function, m@t ksa @x φ = 0, i.e.− φ(x; t) obeys classical wave equation General solutions of the form: φ+(x + vt) + φ−(x vt) p − where v = a ks=m is sound wave velocity and φ± are arbitrary smooth functions φ+ x=νt x=-νt φ– . Comments Low-energy collective excitations { phonons { are lattice vibrations • propagating as sound waves at constant velocity v Trivial behaviour of model is consequence of simplistic definition: • Lagrangian is quadratic in fields linear equation of motion 7! Higher order gradients in expansion (i.e. (@2φ)2) dispersion 7! Higher order terms in potential (i.e. interactions) dissipation 7! L is said to be a `free (i.e. non-interacting) scalar (i.e. one-component) field theory' • In higher dimensions, field has vector components transverse and longintudinal modes • 7! Variational principle is example of functional analysis { useful (but not essential method for this course) { see lecture notes Lecture Notes October 2006 Lecture II 4 Lecture II: Collective Excitations: From Particles to Fields Quantising the Classical Field Having established that the low energy properties of the atomic chain are represented by a free scalar classical field theory, we now turn to the formulation of the quantum system. Canonical Quantisation procedure Recall point particle mechanics: 1. Define canonical momentum, p = @x_ L 2. Construct Hamiltonian, H = px_ L(p; x) − 3. Promote position and momentum to operators with canonical commutation relations x x;^ p p;^ [^p; x^] = i~;H H^ 7! 7! − 7! Natural generalisation to continuous field: @ 2 1. Canonical momentum, π(x) L , i.e. applied to chain, π = @ _ (mφ_ =2) = mφ_ ≡ @φ_(x) φ 2. Classical Hamiltonian Hamiltonian density (φ, π) H Z z }| { 2 h i 1 2 ksa 2 H[φ, π] dx πφ_ (@xφ, φ_) ; i:e: (φ, π) = π + (@xφ) ≡ − L H 2m 2 3. Canonical Quantisation (a) promote φ(x) and π(x) to operators: φ φ^, π π^ 7! 7! 0 0 (b) generalise commutation relations, [^π(x); φ^(x )] = i~δ(x x ) −N.B. [δ(−x x0)] = [Length]−1 (Ex.) − Operator-valued functions φ^ andπ ^ referred to as quantum fields H^ represents a quantum field theoretical formulation of elastic chain, but not yet a solution. As with any function, φ^(x) andπ ^(x) can be expressed as Fourier expansion: φ^(x) 1 X ^ ^ 1 Z L=Na φ^(x) = e±ikx φk ; φk dx e∓ikx π^(x) L1=2 π^ π^ ≡ L1=2 π^(x) k k k 0 P ^ runs over all discrete wavevectors k = 2πm=L; m , Ex: confirm [^πk; φk0 ] = i~δkk0 k 2 Z − Advice: Maintain strict conventions(!) | we will pass freely between real and Fourier space. Lecture Notes October 2006 Lecture II 5 ^y ^ ^y ^ Hermiticity: φ (x) = φ(x), implies φk = φ−k (similarlyπ ^). Using δk+k0;0 z }| { Z L Z L 2 X 0 1 i(k+k0)x X 2 dx (@φ^) = (ikφ^ )(ik φ^ 0 ) dx e = k φ^ φ^ k k L k −k 0 k;k0 0 k 2 m!k=2 z }|2 { X h 1 ksa 2 ^ ^ i 1=2 H^ = π^kπ^−k+ k φkφ−k ;!k = v k ; v = a(ks=m) 2m 2 j j k i.e. `modes k' decoupled Comments: H^ describes low-energy excitations of system (waves) • in terms of microscopic constituents (atoms) However, it would be more desirable to develop picture where • relevant excitations appear as fundamental units: . Quantum Harmonic Oscillator (Revisited) p^2 1 H^ = + m!2q^2 2m 2 Defining ladder operators r r m! i y m! i y 1 a^ x^ + p^ ; a^ x^ p^ ; H^ = ~! a^ a^ + ≡ 2~ m! ≡ 2~ − m! 2 If we find state 0 s.t.a ^ 0 = 0 ; H^ 0 = ~! 0 , i.e. 0 is g.s. j i j i j i 2 j i j i Using commutation relations [^a; a^y] = 1, one may then show n a^yn 0 j i ≡ j i 1 is eigenstate with eigenvalue ~!(n + 2 ) ! Comments: Although single-particle, a-representation suggests many-particle interpretation 0 represents `vacuum', i.e. state with no particles •j i y a^ 0 represents state with single particle of energy ~! • j i a^yn 0 is n-body state, i.e. operatora ^y creates particles • j i Lecture Notes October 2006 Lecture II 6 y 1 y In `diagonal' form H^ = ~!(^a a^ + ) simply counts particles (viz.a ^ a^ n = n n ) • 2 j i j i and assigns an energy ~! to each . Returning to harmonic chain, consider r r m!k ^ i y m!k ^ i ak φk + π^−k ; ak φ−k π^k ≡ 2~ m!k ≡ 2~ − m!k N.B. By convention, drop hat from operators a i~δkk0 z − }| { y i ^ ^ with [ak; ak0 ] = [^π−k; φ−k0 ] [φk; π^k0 ] = δkk0 2~ − . And obtain (Ex. - PS I) X 1 H^ = ! ay a + ~ k k k 2 k Elementary collective excitations of quantum chain (phonons) y created/annihilated by operators ak and ak Spectrum of excitations is linear !k = v k (cf. relativistic) j j Comments: Low-energy excitations of discrete model involve slowly varying collective modes; • i.e. each mode involves many atoms; Low-energy (k 0) long-wavelength excitations, • ! 7! i.e. universal, insensitive to microscopic detail; Allows many different systems to be mapped onto a few classical field theories; • Canonical quantisation procedure for point mechanics generalises to quantum field theory; • Simplest model actions (such as the one considered here) are quadratic in fields • { known as free field theory; More generally, interactions ; non-linear equations of motion viz.
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