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. 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 1. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department ������ ����������� ������� ����� ������ � ������������ �� ��� ���� ����� ����� �������� ��� ���� — ���� ���� ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� ������������ �� ��� ���� ������ ����� �� ��� ����������� � L= where � = 12 ∂ = µφ ∂ µ d3 x φ− � 1 2 2 m φ 2 1 1 2 1 φ̇ − ∇i φ ∇i φ − m2 φ2 2 2 2 ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –1– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 2. �� �������� ������������� L = L(qi , q̇i , t) ∂L ∂q̇i pi ≡ � H= pi q̇i − L i [qi , pj ] = ih̄ δij Schrödinger picture: ih̄ ∂ |ψ(t) = H |ψ(t) . ∂t If H is independent of time, |ψ(t) = e−iHt/h̄ |ψ(0) . ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –2– Heisenberg picture: � (t) = e iHt/h̄ � � � ψ� so that � � � (t)�� ψ �e −iHt/h̄ � � � = ψ (t) � � , � �� ψ(t) � . Relativistic quantum field theory is usually done in the Heisenberg picture: puts space and time on equal footing. ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –3– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 3. �� ����� ������������ �� ������ �������������� Motivation: 1) Can use canonical formulation exactly, instead of as a starting point for a “natural” generalization. Note that δij might naturally generalize to δ 3 (x − x ), but the two expressions do not even have the same units. δij is dimensionless, while δ 3 (x − x ) has units of 1/L3 . 2) For interacting theories, the lattice formulation is the easiest way to understand renormalization. When strong coupling is essential, as in QCD (Quantum Chromodynamics) and the strong interactions, the lattice is even the best way to calculate. ���� ���� ������������ ��������� �� ��������� –4– ������ �������� ��� ���� Start with discrete lattice, spacing = a, and finite volume. Later take limits a → 0 and V → ∞. For the free theory, these limits are trivial. L= � k � k ∆V , where ∆V = a 3 and � 1 1 1 2 φ̇k − ∇i φk ∇i φk − m2 φ2k , 2 2 2 and the lattice derivative is defined by k = ∇i φk ≡ φk� (k,i) − φk a , where k (k, i) denotes the lattice site that is a distance a in the ith direction from the lattice site k . ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –5– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 4. Canonical momenta: pk = ∂L ∂φ̇k = φ̇k ∆V . Define a canonical momentum density: πk ≡ Hamiltonian: H= � � pk = φ̇k . ∆V � pk φ̇k − L = k �� πk φ̇k − k � � ∆V . k Canonical commutation relations: ¯ δk � k . [φk� , φk ] = 0 , [pk� , pk ] = 0 , and [φk� , pk ] = ih In terms of the canonical momentum densities, [φk� , φk ] = 0 , [πk� , πk ] = 0 , and [φk� , πk ] = ih ¯ δk � k . ∆V ���� ���� ������������ ��������� �� ��������� –6– ������ �������� ��� ���� Continuum limit: k πk ≡ � 1 ∂L ∂ k = ∆V ∂φ̇k ∂φ̇k =⇒ =⇒ x π(x , t) = ∂ � ∂φ̇(x , t) = φ̇(x , t) . Hamiltonian becomes H= �� πk φ̇k − k � H= � � d3 x πφ̇ − � k � ∆V =⇒ � . ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –7– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 5. Canonical commutation relations: � � � � φ(x , t) , φ(x , t) = 0 and π(x , t) , π(x , t) = 0 , [φk� , πk ] = ih̄ δk� k ∆V � =⇒ � φ(x , t) , π(x , t) = ih̄ δ (x − x ) , since � � δk � k ∆V = 1 if k ∈ ∆V 0 otherwise k∈ � � � =⇒ x∈ � d3 x δ 3 (x − x ) Note: Dirac delta function is defined by � � 3 d x f (x ) δ (x − x ) ≡ f (x ) x∈� 0 if x ∈ � otherwise. ���� ���� ������������ ��������� �� ��������� –8– ������ �������� ��� ���� �� ������ �� ������ ������� ���������� Choose m ≡ 1, so 1 2 1 2 2 q̇ − ω q , 2 2 ∂L p= = q̇ , ∂q̇ 1 1 H = pq̇ − L = p2 + ω2 q2 . 2 2 Canonical commutation relation: L= [q , p] = ih̄ . Define creation and annihilation operators � � i ω i ω † a= q+ p, a = q− p, 2h̄ 2h̄ω 2h̄ 2h̄ω so that � ���� ���� � a , a† = 1 . ������������ ��������� �� ��������� ������ �������� ��� ���� –9– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 6. The Hamiltonian can be rewritten as � � 1 † H = h̄ω a a + , 2 with eigenstates |n with eigenvalues � � 1 En = n + hω ¯ . 2 Then � � H , a† = hω ¯ a, ¯ a† , [H , a] = −hω which implies that a lowers, and a† raises, the eigenvalues of H by h̄ω. a |0 = 0 , ���� ���� ������������ ��������� �� ��������� –10– ������ �������� ��� ���� Normalized nth excited state: 1 � † �n |n = √ |n , a n! Solve for q and p: � q= � h̄ � a + a† , p = −i 2ω � � hω ¯ � a − a† . 2 In the Heisenberg picture, � q (t) = � h̄ � −iωt ae + a† eiωt 2ω � and p(t) = q̇(t) = −i � hω ¯ � −iωt ae − a† eiωt . 2 ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –11– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 7. �� ������������ �� ��� ����� ������ Continuum limit of the lattice: ∂ π(x , t) = � = φ̇(x , t) , ∂φ̇(x , t) � � � 3 d x πφ̇ − , H= � � � � � φ(x , t) , φ(x , t) = 0 and π(x , t) , π(x , t) = 0 , and � � φ(x , t) , π(x , t) = ih ¯ δ (x − x ) . Use these relations to build the quantum theory. Fourier transform: φ̃(k , t) ≡ and π̃(k , t) ≡ ���� ���� � � d3 x e−ik · x φ(x , t) , d3 x e−ik · x π(x , t) , ������������ ��������� �� ��������� ������ �������� ��� ���� –12– � Fourier inversion: φ(x , t) = � and π(x , t) = d3 k i k · x e φ̃(k , t) (2π)3 d3 k i k · x e π̃ (k , t) . (2π)3 φ(x , t) real =⇒ Hermitian quantum operator =⇒ φ̃(−k , t) = φ† (k , t) , π̃ (−k , t) = π† (k , t) . Heisenberg equations of motion: ∂ 2φ 2 φ + m2 φ = 0 . −∇ ∂t2 ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –13– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 8. Since h̄ = 1, m has the units of an inverse length, not a mass. Fourier transform obeys ∂ 2 φ̃ (k , t) + (k 2 + m2 ) φ̃(k , t) = 0 . ∂t2 General solution: φ̃(k , t) = φ1 (k )e−iωp t + φ2 (k )eiωp t , � ωp = k 2 + m2 . where Reality φ̃(−k , t) = φ† (k , t) implies φ2 (k ) = φ†1 (−k ) , so φ̃(k , t) = φ1 (k )e−iωp t + φ†1 (−k )eiωp t . Since π = φ̇, π̃ (k , t) = −iωp φ1 (k )e−iωp t + iωp φ†1 (−k )eiωp t . ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –14– These two equations can be solved simultaneously to give: � � i 1 φ1 (k ) = φ̃(k , t) + π̃ (k , t) eiωp t ωp 2 � � � i 1 3 −i( x −ωp t) k · = φ(x , t) + π(x , t) , d xe ωp 2 Using canonical commutation relations, � � � 1� 3 φ1 (k ) , φ1 (q ) = d3 y e−i(k · x −ωk t) e−i( q · y −ωq t) dx 4 � � i i π(x , t) , φ(y , t) + π(y , t) × φ(x , t) + ωk ωq � � 1 3 d3 y e−i(k · x −ωk t) e−i( q · y −ωq t) dx = 4 � � i i 3 3 × ih ¯ δ (x − y ) − ih ¯ δ (x − y ) ωq ωk = ... . –15– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 9. Using canonical commutation relations, � � � 1� 3 d3 y e−i(k · x −ωk t) e−i( q · y −ωq t) φ1 (k ) , φ1 (q ) = dx 4 � � i i × φ(x , t) + π(x , t) , φ(y , t) + π(y , t) ωk ωq � � 1 3 = d3 y e−i(k · x −ωk t) e−i( q · y −ωq t) dx 4 � � i i 3 3 × ih̄ δ (x − y ) − ih̄ δ (x − y ) ωq ωk � � � �� 1 h̄ 1 −i ( q )· x −(ωk +ωq )t k + 3 d xe − . = 4 ωk ωq ���� ���� ������������ ��������� �� ��������� –16– ������ �������� ��� ���� � But d3 x e−i(k + q )· x = (2π)3 δ 3 (k + q ) , which requires q = −k , but then the factor in curly brackets vanishes. Thus, � � φ1 (k ) , φ1 (q ) = 0 , as expected if φ1 (k ) is proportional to an annihilation operator. ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –17– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 10. Similarly, � � �� � h̄ � � 1 1 † −i ( x −(ωk −ωq )t q )· k − 3 φ1 (k ) , φ1 (q ) = + . d xe 4 ωk ωq = h̄ h̄ 2 (2π)3 δ 3 (k − q ) = (2π)3 δ 3 (k − q ) . 4 ωk 2ωk Continuum normalization convention: � Define a(k ) = 2ωk φ1 (k ) . h̄ so that � � � � a(k ) , a(q ) = 0 , a(k ) , a† (q ) = (2π)3 δ 3 (k − q ) , ���� ���� ������������ ��������� �� ��������� –18– ������ �������� ��� ���� Then � φ(x , t) = d3 k (2π)3 � � h̄ � i( k ·x−ωk t) a( k ) e + a† (k )e−i(k · x −ωk t) . 2ωk Note: in 2nd term, I changed variables of integration k → −k , Canonical momentum density: π(x , t) = φ̇(x , t) � � � d3 k h̄ωk � i( k ·x−ωk t) † −i( x −ωk t) k · = −i a ( ) e − a ( ) e . k k (2π)3 2 ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –19– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 11. Creation and annihilation operators: a(k ) = � a (k ) = † ωk 2h̄ � � ωk 2h̄ −i( x −ωp t) k · d xe 3 � i( x −ωp t) k · d xe 3 � � i φ(x , t) + π(x , t) . ωp � i φ(x , t) − π(x , t) ωp � . ���� ���� ������������ ��������� �� ��������� ������ �������� ��� ���� –20–