From Particles to Fields

Transcription

1 From Particles to Fields Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 25, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

2 Hamiltonian A Hamiltonian of a prototypical metal or insulator is given by: where H = H e + H i + H ei H e = i H i = I H ei = ii p 2 i 2m + V ee (r i r j ), i<j P 2 I 2M + V ii (R i R j ), i<j V ei (R I r i ) Here, we denote r i the coordinates of the valence electrons, R I the coordinates of the ion cores. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

3 How to simplify the model: It is difficult to make theoretical progress by an approach that treats all microscopic constituents as equally relevant degrees of freedom. How can successful analytical approaches be developed? Structural reducibility: Not all components of the Hamiltonian need to be treated simultaneously. Universality: In the majority of condensed matter application, one is interested not so much in the full profile of a given system, but rather in its energetically low-lying dynamics. At low temperature, large systems tends to behave in a universal manner. Concepts of statistics: For most system of interest, the degree of freedom is very large (N = O(10 23 )). Statistical error tend to be negligibly small as N. Symmetries: A condensed matter observable is generally tied to an energetically low-lying excitation. These low-level excitations governs the gross behavior of the system. symmetry conservation law low-lying excitations. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

4 Classical harmonic chain: phonons Here, we consider the dynamical properties of the positively core ions that constitute the host lattice of a crystal. For a moment, we treat the ions as classical object instead of quantum ones. We assume that the dynamics of electrons are secondary importance in considering the behavior of the ions. So we set H e = H el = 0. The reduced low-energy Hamiltonian of the system is given by H = N I =1 [ P 2 I 2M + k ] s 2 (R I R I +1 a) 2 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

5 Classical harmonic chain: the discrete Lagrangian The Lagrangian action S = t 0 0 [ N MṘI 2 L = T U = 2 I =1 L(R, Ṙ) dt where k s 2 (R I R I +1 a) Consider a large-n system with a circle topology, i.e. R N+1 = R 1. Here, we assume the deviation of the ions from equilibrium is small ( R I (t) R I a). The Lagrangian could be expressed as L = N I =1 where φ I (t) = R I (t) R I. [ M 2 φ 2 I k ] s 2 (φ I +1 φ I ) 2 ]. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

6 Classical harmonic chain: the Lagrangian density Since we are interested in experimental behavior which manifests itself on macroscopic length scales, we take continuum limit for the discrete Lagrangian. Using φ I a 1/2 φ(x) x=ia, φ I +1 φ I a 3/2 x φ(x) x=ia, where L = Na, we obtain the Lagrangian density N 1 a I =1 L 0 dx. L[φ] = L 0 L(φ, x φ, φ) dx, L(φ, x φ, φ) = m 2 φ 2 k sa 2 The continuum form of the classical action is L S[φ] = dt L[φ] = dt dx L(φ, x φ, φ). 0 2 ( xφ) 2. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

7 Classical harmonic chain: Hamilton s principle Suppose the dynamics of a classical point particle with coordinate x(t) is described by the classical Lagrangian L(x, ẋ), and action S[x] = dt L(x, ẋ). The Hamilton s principle states that the configuration x(t) of that are actually realized are those that extremize the action, δs[x] = 0. We obtain the Lagrange s equation of motion d dt ( ẋl) x L = 0. Now we apply the Hamilton s principle to a system of infinitely many degree freedom, φ(x, t), we obtain the equation of motion which take the form of a wave equation (m 2 t k s a 2 2 x )φ = 0. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

8 Classical harmonic chain: Collective excitations The general solution to the wave equation φ + (x vt) + φ (x + vt) where v = a k s /m and φ ± are arbitrary smooth functions. The basic low-energy elementary excitations of the model are lattice vibrations propagating as sound waves. The elementary excitations of the chain have little in common with its microscopic constituents (the atomic oscillators). They are collective excitation, i.e. elementary excitations + a microscopically large number of microscopic degrees of freedom. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

9 Classical harmonic chain: Hamiltonian formulation Define the canonical momentum as π(x) L(φ, xφ, φ). φ(x) The Hamiltonian density is given by H(φ, x φ, π) = (π φ L(φ, x φ, φ)) φ= φ(φ,π) where the full Hamiltonian H = L 0 H dx. The full Hamiltonian is ( π 2 H[π, φ] = dx 2m + k sa 2 ) 2 ( xφ) 2. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

10 Classical harmonic chain: the specific heat The thermodynamic energy density of the microscopic harmonic oscillators is u = 1 dγe βh H L dγe βh = 1 L β ln dγe βh where β = 1 k B T and the Boltzmann partition function Z dγe βh and the phase space volume element dγ = N I =1 dr I dp I. After rescaling R I β 1/2 X I, P I β 1/2 Y I, βh(r, P) H(X, Y ), we obtain u = 1 L β ln(β N K) = ρt where ρ = N/L is the density of the atoms, and K dγ e H(X,Y ) is a constant independent of T. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

11 Comparison the specific heat with experimental results The specific heat is c = T u = ρ. Note that c is independent of the material constant M and k s. The energy of a system with N degrees of freedom is U = Nk B T. For a large temperature, the specific heat approaches to a constant. For low temperature, the deviation of specific heats is due to a quantum phenomenon. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

12 Maxwell s equations Maxwell s equations: E = ρ B t E = J E t B = 0 B = 0 Coulomb s law Ampere s law Faraday s law Absence of free magnetic poles Let A µ = (φ, A) be the electromagnetic 4-potential satisfied E = 1 A c t Φ B = A For arbitrary function Γ, a gauge transformation is given by A µ A µ + µ Γ The EM field is invariant under the gauge transformation. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

13 Expressions of physical fields The relation between fields and potential is F µν = µ A ν ν A µ where x µ = (t, x) and µ = ( t, ). In matrix form, we have The dual tensor is F = {F µν } = F = {F µν } = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

14 Lorentz transformation and Lorentz invariance Set g = {g µν } = A linear coordinate transformation X µ X µ T µν X ν is called a Lorentz transformation if T t gt = g. Defining X µ g µν X ν, Lorentz invariance is expressed as X µ X µ = X µ X µ. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

15 Lagrangian formulation of electrodynamics The Lagrangian of electromagnetic field: L(A µ, ν A µ ) = ( 1 4 F µνf µν + A µ j µ ) The corresponding action is S[A] = d 4 x L(A µ, ν A µ ). This action is invariant under Lorentz transformations and gauge transformations. The equations of motion of the electromagnetic field are ν F νµ = j µ which is nonhomogeneous part of the Maxwell s equation. The homogeneous parts of the Maxwell equations are satisfied automatically. ν F νµ = 0. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

16 From Hamilton s equation to quantum mechanics The Poisson bracket is defined as [ N f g {f, g} = f ] g q i=1 i p i p i q i Let (q i, p i ) be canonical coordinates in phase space and f be some function of p, q and H. {q i, q j } = 0, {p i, p j } = 0, {q i, p j } = δ ij. The general form of Hamilton s equation can be expressed as df dt = {f, H} + f t. The Hamilton s equation can be written as q i = {q i, H} = H p i p i = {p i, H} = H q i The commutator of two operators A, B is defined by [A, B] = AB BA. The canonical quantization prescription: {f, g} 1 i [ˆf, ĝ] Without confusion, we will dropˆ. The canonical quantization transform the general Hamilton s equation to Heisenberg s picture of quantum mechanics d ˆF dt (t) = i [Ĥ, ˆF (t)] + ˆF t where Ĥ is the Hamiltonian, ˆF (t) is some observable. The Heisenberg equation of motion is equivalent to the Schrodinger equation Ĥ Ψ(t) = i t Ψ(t). ien-tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

17 Quantum chain At low temperature, the excitation profile of the classical atomic chain differs drastically from observations in experiments. Low-energy phenomena with pronounced temperature sensitivity are generally indicative of a quantum mechanism. Now we consider the low-energy physics of the quantum mechanical chain. Ĥ( ˆφ, ˆπ) = 1 2m ˆπ2 + k sa 2 2 ( x ˆφ) 2 where ˆφ and ˆπ are operator-valued functions, referred to as quantum fields. Relations between discrete and continuum canonically conjugate variables and operators. Classical Quantum Discrete {p i, q j } = δ ij [ˆp i, ˆq j ] = i δ ij Continuum {π(x), φ(x )} = δ(x x ) [ˆπ(x), ˆφ(x )] = i δ(x x ) Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

18 Quantum chain: Fourier representation Fourier representation of ˆφ(x) and ˆπ(x): { ˆφ k 1 L { ˆφ dxe ikx ˆπ k L 1/2 ˆπ, 0 where quantized momenta k = 2πm/L for all m Z. { ˆφ ˆπ 1 L 1/2 The Fourier representation of the canonical commutation relations [ˆπ k, ˆφ k ] = δ kk. k dxe ±ikx { ˆφk ˆπ k Hermitian quantum field of ˆφ(x) and ˆπ(x) imply ˆφ k = ˆφ k and ˆπ k = ˆπ k. In the Fourier representation, we have dx( ˆφ) 2 = ( ik ˆφ k )( ik ˆφk ) 1 dx e i(k+k )x = L k,k k k 2 ˆφk ˆφ k = k k 2 ˆφ k 2. The Hamiltonian Ĥ = dxh( ˆφ, ˆπ) takes the form Ĥ = [ ] 1 2m ˆπ k ˆπ k + mω2 k 2 ˆφ k ˆφ k k where ω k = v k and v = a k s/m denotes the classical sound wave velocity. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

19 Review of the quantum harmonic oscillator Consider a standard harmonic oscillator Hamiltonian Ĥ = ˆp2 2m + mω2 2 ˆx 2. The energy level is ɛ n = ω(n + 1 ) and the corresponding eigenstate is the 2 associated Hermite polynomial. Equidistant of the energy level suggests an alternative interpretation. We could think of the given energy state ɛ n as an accumulation of n elementary entities or quasi-particles, each having energy ω. Define the annihilation operator and the creation operator mω â 2 (ˆx + i mω mω ˆp), â 2 (ˆx i mω ˆp). The Hamiltonian can be expressed as Ĥ = ω ( â â + 1 ). 2 We have â 0 = 0, H 0 = ω 2 0, n 1 n! (â ) n 0 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

20 Quasi-particle interpretation of the quantum chain Define the ladder operators ( mωk â k = ˆφ k + 2 i mω k ˆπ k ) (, â k = mωk ˆφ k 2 They have the following commutation relations: [a k, a k ] = δ kk, [a k, a k ] = [a k, a k ] = 0. The Hamiltonian of the quantum chain becomes Ĥ = ( ω k a k a k + 1 ) 2 k i ) ˆπ k. mω k which is the sum of harmonic oscillator with characteristic frequency ω k. The excited state of the system is indexed by a set {n k } = (n 1, n 2,... ) of quasi-particle with energy ω k. The quasi-particle of the harmonic chain are identified with the phonon modes of the solid. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

21 Quantum electrodynamics (I) Consider the Lagrangian of the matter-free EM field, L = 1 4 electrodynamic waveguide. d 3 x F µνf µν in an Using the Coulomb gauge, A = 0, with φ = 0. The Lagrangian assume the form L = 1 dx 3 [( ta) 2 ( A) 2 ]. 2 The equation of motion for the EM wave is The eigenvalue equation is tta 2 A = 0. 2 R k (x) = λ k R k (x). We could choose the following real orthonormal vector-valued functions R k = N k c 1 cos(k xx) sin(k y y) sin(k zz) c 2 sin(k xx) cos(k y y) sin(k zz) c 3 sin(k xx) sin(k y y) cos(k zz) where k i = n i π/l i, n i Z, i = x, y, z, and N is normalizing factor and c i are constant coefficients with c 1k x + c 2k y + c 3k z = 0. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

22 Quantum electrodynamics (II) Quantize the theory by α k ˆα k and π k ˆπ k. We set ω 2 k = λ k. The quantum Hamiltonian operator is Suppose A(x, t) can be expanded in R k. It is A(x, t) = k α k (t)r k (x). Ĥ = k ( 1 2 ˆπ k ˆπ k + ω ) 2 k 2 ˆα k ˆα k where k = πn/l, n Z and L is the length of the waveguide. The classical Lagrangian can be expressed as L = 1 ( α 2 k 2 λ k α 2 k ). k Define the ladder operator ωk a k 2 ˆα k + i ˆπ k ω k a k ωk 2 ˆα k i ω k ˆπ k. Set the momenta π k = αk L = α k. The classical Hamiltonian is H = k ( 1 2 π k π k + λ k 2 α k α k ). Quantize the theory by α k ˆα k and π k ˆπ k. The quantum Hamiltonian operator is The quantum Hamiltonian operator can rewrite as Ĥ = k ( ω k a k a k + 1 ). 2 The construction above is almost paralleled to the discussion of the harmonic chain. The quantum excitations are the photons of the EM waves. Ĥ = ( 1 2 ˆπ k ˆπ k + λ ) k 2 ˆα k ˆα k k ien-tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

23 Phonon specific heat for quantum chain The quantum partition function Z = tr e βĥ e βestate = e β m ωm(nm+1/2) = states m m=1,2,... n m=0 m=1,2,... e βωm/2 1 e βωm. The energy density of one-dimensional longitudinal phonons with dispersion ω k = v k is u = 1 L β ln Z = 1 L [ ] β β ωm 2 + ln(1 e βωm ) = 1 ( ) 1 ω k L 2 + n B(ω k ). where The Bose-Einstein distribution function is n B (ɛ) = 1 e βɛ 1. Replacing m L 2π dk, we have u = C v k C2 dk = C1 + 2π eβv k 1 β 2 k <Λ where the Hamiltonian have a nice quadratic approximation for k < Λ for some cut-out momentum Λ. For low temperature T, we have c v = T u T. k Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24

24 Symmetry Noether s theorem states that any differentiable symmetry of a physical system has a corresponding conservation law. Lagrangian mechanics: Suppose the Lagrangian is unchanged under q i q i + δq i. Because of L q i = 0, we have dp i dt = d ( ) L = 0 dt q i Hamiltonian mechanics: Suppose H = H(q i, p i ). We have dp i dt = 0 H whenever = 0. q i Quantum mechanics: Let T (ɛ) = exp( iɛ G) be a unitary operator with a generator G. Suppose H is invariant under T (ɛ). i.e. T HT = H or [G, H] = 0. Therefore we have dg dt = 0. Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, / 24