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ft1ls01p1_08

# ft1ls01p1_08 - MIT OpenCourseWare http/ocw.mit.edu 8.323...

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Unformatted text preview: 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 � = 1∂ 2 = µφ ∂ µ d3 x φ− � 1 22 mφ 2 1 1 ˙2 1 φ − ∇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 , qi , t) ˙ ∂L ∂ qi ˙ pi ≡ � H= pi qi − L ˙ i ¯ [qi , pj ] = ih δij Schr¨dinger picture: o ih ¯ ∂ |ψ(t) = H |ψ(t) . ∂t If H is independent of time, ¯ |ψ(t) = e−iH t/h |ψ(0) . ����� ������� ��������� �� �� ������� –2– ���� ���� ������ �������� ��� ���� Heisenberg picture: � (t) = e iH t/h ¯ �� � ψ� so that �� � (t)�� ψ �e −iH t/h ¯ � � � = ψ (t) � � , � �� ψ(t) � . Relativistic quantum ﬁeld theory is usually done in the Heisenberg picture: puts space and time on equal fo oting. ���� ���� ����� ������� ��������� �� �� ������� ������ �������� ��� ���� –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 φk − ∇i φk ∇i φk − m2 φ2 , k 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 ˙ = φk ∆V . ˙ ∂ φk Deﬁne a canonical momentum density: πk ≡ Hamiltonian: H= � � pk ˙ = φk . ∆V � ˙ pk φk − L = k �� ˙ πk φk − k � � k ∆V . Canonical commutation relations: ¯ [φk� , φk ] = 0 , [pk� , pk ] = 0 , and [φk� , pk ] = ih δk� k . 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 = ˙k ˙ ∆V ∂ φ ∂ φk =⇒ =⇒ x π (x , t) = � ∂ ˙ = φ(x , t) . ˙ (x , t) ∂φ Hamiltonian becomes H= �� ˙ πk φk − k � H= � � ˙ d3 x πφ − � 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 ] = ih δk� k ¯ ∆V � =⇒ � φ(x , t) , π(x , t) = ih 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– ������ �������� ��� ���� �� ������ �� ������ ������� �� �������� Cho ose m ≡ 1, so 12 1 22 ˙ q− ωq, 2 2 ∂L p= =q , ˙ ∂q ˙ 1 1 H = pq − L = p2 + ω2 q2 . ˙ 2 2 Canonical commutation relation: L= ¯ [q , p] = ih . Deﬁne creation and annihilation operators � � i ω i ω † a= q+ p, a = q− p, 2¯ h 2hω ¯ 2h ¯ 2hω ¯ 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 = hω a a + ¯ , 2 with eigenstates |n with eigenvalues � � 1 En = n + hω . ¯ 2 Then � � H , a† = hω a† , [H , a] = −hω a , ¯ ¯ which implies that a lowers, and a† raises, the eigenvalues of H by hω. ¯ a |0 = 0 , ���� ���� ����� ������� ��������� �� �� ������� –10– ������ �������� ��� ���� Normalized nth excited state: 1 � † �n |n = √ |n , a n! Solve for q and p: � q= � h� ¯ a + a† , p = −i 2ω � � hω � ¯ a − a† . 2 In the Heisenberg picture, � q (t) = � h � −iωt ¯ ae + a† eiω t 2ω � and p(t) = q(t) = −i ˙ � hω � −iωt ¯ ae − a† eiω t . 2 ���� ���� ����� ������� ��������� �� �� ������� ������ �������� ��� ���� –11– 8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 7. �� ������������ �� ��� � ���� ������ Continuum limit of the lattice: � ∂ ˙ = φ (x , t) , ˙ ∂ φ(x , t) π (x , t) = � � � ˙− d x πφ , H= � � � � φ(x , t) , φ(x , t) = 0 and π(x , t) , π(x , t) = 0 , 3 and � � � φ(x , t) , π(x , t) = ih δ (x − x ) . ¯ Use these relations to build the quantum theory. � Fourier transform: ˜ φ(k , t) ≡ � and ˜ π (k , t) ≡ d3 x e−ik ·x π(x , t) , ����� ������� ��������� �� �� ������� –12– ������ �������� ��� ���� ���� ���� d3 x e−ik ·x φ(x , t) , � Fourier inversion: φ(x , t) = � and π (x , t) = d3 k ik ·x ˜ e φ(k , t) ( 2π )3 d3 k ik ·x e π (k , t) . ˜ ( 2π )3 φ(x , t) real =⇒ Hermitian quantum operator =⇒ ˜ φ(−k , t) = φ† (k , t) , π (−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 h = 1, m has the units of an inverse length, not a mass. Fourier transform ¯ obeys ˜ ∂ 2φ ˜ ( k , t ) + ( k 2 + m2 ) φ ( k , t ) = 0 . ∂ t2 General solution: ˜ φ(k , t) = φ1 (k )e−iωp t + φ2 (k )eiωp t , � ωp = k 2 + m2 . where ˜ Reality φ(−k , t) = φ† (k , t) implies φ2 (k ) = φ† (−k ) , 1 so ˜ φ(k , t) = φ1 (k )e−iωp t + φ† (−k )eiωp t . 1 ˙ Since π = φ, π (k , t) = −iωp φ1 (k )e−iωp t + iωp φ† (−k )eiωp t . ˜ 1 ���� ���� ����� ������� ��������� �� �� ������� ������ �������� ��� ���� –14– These two equations can be solved simultaneously to give: � � i 1˜ φ1 (k ) = φ(k , t) + π (k , t) eiωp t ˜ ωp 2 � � � i 1 3 −i(k ·x −ωp t) = 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 × ih δ (x − y ) − ¯ ih δ (x − y ) ¯ ωq ωk � � � �� 1 h 1 ¯ −i (k +q )·x −(ωk +ωq )t 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, � � �� � h� � 1 ¯ 1 † −i (k −q )·x −(ωk −ωq )t 3 φ1 (k ) , φ1 (q ) = + . d xe 4 ωk ωq = h h ¯ ¯ 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 ) . h ¯ so that � � � � a( k ) , a( q ) = 0 , a( k ) , a† ( q ) = ( 2 π ) 3 δ 3 ( k − q ) , ���� ���� ����� ������� ��������� �� �� ������� –18– ������ �������� ��� ���� Then � φ(x , t) = d3 k ( 2π )3 � � h� ¯ a(k ) ei(k ·x−ωk t) + 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) = φ (x , t) � � � d3 k hωk � ¯ = −i a(k ) ei(k ·x−ωk t) − a† (k )e−i(k ·x −ωk t) . ( 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 2h ¯ � � ωk 2h ¯ −i(k ·x −ωp t) d xe 3 � i(k ·x −ωp t) d xe 3 � � i φ(x , t) + π (x , t) . ωp � i φ(x , t) − π (x , t) ωp � . ���� ���� ����� ������� ��������� �� �� ������� ������ �������� ��� ���� –20– ...
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