ft1ls06p_08_6a

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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. Alan Guth, 8.323 Lecture Slides 6A, May 8, 2008 — Intro duction Only, p. 1. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department ������ ����������� ������� ����� ������ � ��� ���� ���������� �� ��� ���� ����� ��� ��� ������ ���� ������� ��������� ��� �� ���� — ���� ���� ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ����� �� ���� ������ �� ������ �� ���� ���������� ��������� ��� ��� ���� ����� The Dirac Lagrangian was given as Eq. (120): � Dirac ¯ = ψ iγ µ ∂µ − m ψ , (120) where † 0 ¯ ψa (x) = ψb (x) γba ¯ ψ ( x) = ψ † ( x) γ 0 . (121) The canonical momentum is then π= � ∂ = iψ † , ˙ ∂ψ ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ����� �� ���� ������ �� –1– Alan Guth, 8.323 Lecture Slides 6A, May 8, 2008 — Intro duction Only, p. 2. so we might expect canonical anticommutation relations of the form {ψa (x , 0) , πb (y , 0)} = iδ (3) (x − y ) δab =⇒ † ψa (x , 0) , ψb (y , 0) = δ (3) (x − y ) δab Remember Eq. (99): ¯ ψ (x ,0) , ψ (y ,0) = Ep d3 p 1 ip ·(x −y ) e × ( 2π )3 2E p ∗ m (1 − βL βR ) 1 + |βR |2 + p · σ 1 − |βR |2 Ep 1 + |βL |2 − p · σ 1 − |βL |2 , ∗ m (1 − βR βL ) which for βL = βR = 1 implies that ¯ ψ (x ,0) , ψ (y ,0) = d3 p ip ·(x −y ) 01 = δ (3) (x − y ) γ 0 , e 10 ( 2π )3 ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ����� �� ���� ������ �� –2– which is exactly right. The full canonical anticommutation relations are † ψa (x , 0) , ψb (y , 0) = δ (3) (x − y ) δab † † {ψa (x , 0) , ψb (y , 0)} = ψa (x , 0) , ψb (y , 0) = 0 . ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ����� �� ���� ������ �� –3– Alan Guth, 8.323 Lecture Slides 6A, May 8, 2008 — Intro duction Only, p. 3. H= d3 p Ep (2π )3 ���� ���� ������ a† (p ) as (p ) + b† (p ) bs (p ) + Evac , s s s (134) where Evac = −2 d3 p Ep δ (3) (0) = −2 d3 p Ep × Volume of space . (2π )3 (135) Note that Fermi statistics caused the antiparticle energy to be positive (go o d!), and the vacuum energy to be negative (surprising?). The negative vacuum energy, although ill-defined, is still welcome: allows at least the hope that one might get the positive (bosonic) contributions to cancel against the negative (fermionic) contributions, giving an answer that is finite and hopefully small. Note that if we had 4 free scalar fields with the same mass, the cancelation would be exact: this is what happens in EXACTLY supersymmetric mo dels, but it is spoiled as so on as the supersymmetry is broken. –4– ���� ���� ������ Unoccupied states E Electron +mc2 +mc2 0 Radiation 0 Electron Radiation -mc2 -mc2 Occupied states E Hole Hole Electron-positron pair creation Electron-positron pair annihilation Figure by MIT OpenCourseWare. Adapted from Bjorken & Drell, vol. 1, p. 65. In the 1-particle quantum mechanics formulation, positrons show up as negative energy states. Dirac proposed that in the vacuum, the negative energy “sea” was filled. Physical positrons, in this view, are holes in the Dirac sea. In QFT, on the other hand, particles and antiparticles are on equal footing. Nonetheless, the Dirac sea allows an intuitive way to understand the negative vacuum energy. From Bjorken & Drell, vol. 1, p. 65 –5– Alan Guth, 8.323 Lecture Slides 6A, May 8, 2008 — Intro duction Only, p. 4. �� ��� ������ In quantum field theory (no gravity), vacuum energy is meaningless and can be dropped. In “semiclassical gravity,” in which the expectation value of the energymomentum tensor is taken as the source of a classical gravitational field, the vacuum energy matters, but it can be subtracted. The subtraction, however, does not appear to be theoretically well-motivated. In string theory, any subtraction would destroy the consistency of the theory. The vacuum energy density is exactly zero in supersymmetric vacua, but of h order the Planck scale (ρ ∼ G−2 with ¯ = c = 1) for typical vacua, which is about 120 orders of magnitude too large. There are believed to be maybe 10500 different vacua (the “landscape” of string theory), and anthropic arguments are sometimes used to explain why we find ourselves in one of the unusual vacua with a very small but nonzero vacuum energy. ����� ������� ��������� �� �� ������� –6– ��� ���� ���������� ���� ���� ������ ��� �� ����� �� ���� ������ �� This is straightforward, so I will only summarize the results. ¯ 0 ψ a ( x) ψ b ( y ) 0 = d3 p 1 ( 2π )3 2E p = (i ∂ x + m)ab s us (p ) us (p ) e−ip·(x−y) ¯b a d3 p 1 −ip·(x−y) e ( 2π )3 2E p = (i ∂ x + m)ab D(x − y ) ¯ 0 ψ b ( x) ψ a ( y ) 0 = d3 p 1 ( 2π )3 2E p s (136) s va (p ) vb (p ) e−ip·(y−x) ¯s = −(i ∂ x + m)ab D(y − x) , where ∂ = γ µ ∂µ and D(x) is the scalar 2-point function 0 |φ(x)φ(0)| 0 . ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ����� �� ���� ������ �� –7– Alan Guth, 8.323 Lecture Slides 6A, May 8, 2008 — Intro duction Only, p. 5. ��� �������� ���� ���������� ab ¯ SR (x − y ) ≡ θ (x0 − y 0 ) 0 ψa (x) ψb (y ) 0 (137) = (i ∂ x + m) DR (x − y ) , where DR (x − y ) is the scalar retarded propagator. One can show (i ∂ x − m)SR (x − y ) = iδ (4) (x − y ) · 14×4 . (138) The Fourier expansion is SR (x) = d4 p −ip·x ˜ i( p + m) ˜ SR (p) , where SR (p) = 0 e . 4 (2π ) (p + i )2 − |p |2 − m2 (139) ���� ���� ����� ������� ��������� �� �� ������� –8– ������ ��� �� ����� �� ���� ������ �� SF (x − y ) ≡ ¯ 0 ψ ( x) ψ ( y ) 0 ¯ − 0 ψ ( y ) ψ ( x) 0 ¯ ≡ 0 T ψ ( x) ψ ( y ) ��� ������� ���������� for x0 > y 0 for y 0 > x0 (140) 0. The Feynman propagator also satisfies Eq. (138). The Fourier expansion is SF (x) = d4 p −ip·x ˜ i( p + m) ˜ . SF (p) , where SF (p) = 2 e 4 ( 2π ) p − m2 + i (141) This differs from the scalar field Feynman propagator by the factor ( p + m). ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ����� �� ���� ������ �� –9– Alan Guth, 8.323 Lecture Slides 6A, May 8, 2008 — Intro duction Only, p. 6. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department ������ ����������� ������� ����� ������ � ��� ���� ���������� �� ��� ���� ����� �� �������� ���� ��� �� ���� ������ � ��������� ��� �� ���� — ���� ���� ����� ������� ��������� �� �� ������� ������ ��� �� ����� �� ���� ������ �� ���� ���� ...
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