The Path Integral: Basics and Tricks (largely from Zee) Yichen Shi Michaelmas 03 Path-Integral Derivation x f, t f x i, t i x f e H(t f t i) x i. If we chop the path into N intervals of length ɛ, then using the completeness relation d 3 x x x, we can write N x f, t f x i, t i d 3 x x f e Hɛ x N x N e Hɛ x N x e Hɛ x i. Now, using the completeness relation we can write, for a typical term, k x e Hɛ x d 3 p (π) 3 p p, d 3 p (π) 3 x p p e Hɛ x, x p e p x, and for H p m + V (x), p e Hɛ x e ɛ p m +V (x) p x e ɛ p m +V (x) e ip x. x e Hɛ x If we put things together now, we ll have x f, t f x i, t i N k e ɛ d 3 x k N l p N : d 3 p ep (x x) (π) 3 e ɛ p m +V (x) d 3 p (π) 3 eɛ d 3 p (π) 3 eɛ p x x ɛ + p m +V (x) p m. +V (x)+p ẋ d 3 p p l eɛ f m +V (x f )+p f ẋ N (π) 3 m +V (x N )+p N ẋ N e ɛ p m +V (x)+p ẋi.
But we can first let p : p mẋ, which gives Hence p m + p ẋ p m m p mẋ + mẋ + p ẋ mẋ p m mẋ. x e Hɛ x And now putting things together, we have d 3 p p (π) 3 eɛ m e iɛ( mẋ V (x)) πm (π) 3 iɛ m 3/ e iɛl(x,ẋ). πiɛ 3/ e iɛ( mẋ V (x)) m 3N/ N x f, t f x i, t i lim dx k e is(x f,x i) ɛ 0 πiɛ k. Separating out the Classical Path Define a path x(t) x cl (t) + y(t) x cl (t) x i + t t i t f t i (x f x i ) ẋ cl (t) x f x i t f t i ẍ cl (t) 0, and y(t) is the quantum part, satisfying y(t i ) y(t f ) 0. Then, S m tf m tf t i dt ẋ t i dt ( ẋ cl + ẋ cl ẏ + ẏ ) m x clẋ cl + ẋ cl y tt f tt i m(x f x i ) x f x i t f t i m tf + m tf t i dt ( x cl ẍ cl + ẍ cl y ẏ ) t i dt ẏ x f, t f x i, t i C e i m (x f x i) t f t i e i m t f t dt N (y k+ y k ) i k ɛ. Basic Identities and Quantum Mechanics. The Delta Function (x a) f(x) dk π eik(x a) da (x a)f(a)
(f(x)) x i, s.t.f(x i)0 dx (f(x))s(x) i (x x i ) f (x i ) s(x i ) f (x i ). Gaussian Integrals Using completing the square, dx e a(x+b) dx e ax +Jx dx e ax +ijx dx e i ax +ijx π a π π a ej /a a e J /a πi a ej /a.3 Determinant Integrand.4 ζ Function Regularisation dx e i (πi) x A x det A dx dx N e i x A x+ij x (πi) N det A e i J A J First note that and ζ(0) /, ζ (0) log(π)/. ζ(s) : n Z + n s Now, for Ax λ n x, given λ n a n, we have det A n Z + a n since the determinant of a matrix is the product of its eigenvalues. Then, log(det A) log(a ) + n n Z + n Z + log(a) e 0 d ds s0 n Z + log(a)ζ(0) ζ (0) log(a ) + log(π) log(π/a). n Z + exp s log(n) 3
.5 Rewriting Terms as Derivatives Expanding in λ first, Z(λ, J) Z(0, 0) π/m. λ 4! dq e m q +Jq q 4n ( d ) 4n dq e m q +Jq dj ( d ) n dm dq e m q +Jq. dq e m q λ 4! q4 +Jq dq e m q +Jq λ 4! q4 + ( λ 4! ) 8 + ( d dj e λ 4! ( d dj )4 ) 4 + Z(0, 0) e λ 4! ( d dj )4 e J m Z(0, 0) λ ( d 4! dj ( λ ) ( d 4! dj dq e m (q J m ) + J m ) 4 + ) q 8 + dq e m q +Jq ( λ ) ( d ) 8 + + J 4! dj m + ( J ) + m, dq dq N e q A q λ 4! q4 +J q (π) N λ deta e 4! k ( J ) 4 k e J A J..6 Wick Contracion From the above, alternatively, expanding in J first, Z(J) s0 dq e m q λ 4! q4 +Jq s! J s : Z(0, 0) + s0 dq e m q λ 4! q4 q s s! J s G (s). s0 i Z(0, 0) dq dq N e q A q λ 4! q4 +J q N N + ( ) s! J i J is dql e q A q (λ/4!)q4 q i q is i s s0 i N N i s s! J i J is G (s) i i s. 4
.7 n-point Green s Function -point Green s function to zeroth order in λ is given by ( ) G () ij (λ 0) dq l e q A q q i q j /Z(0, 0) : (A ) ij. 4-point Green s function to first order in λ is given by ( ) G (4) ij dq m e q A q q i q j q k q l ( λ ) 4! (q m) 4 Z(0, 0) m l (A ) ij (A ) kl + (A ) ik (A ) jl + (A ) il (A ) jk λ n (A ) in (A ) jn (A ) kn (A ) ln..8 Feynman Rules from the Path-integral Z(J) corresponds to Feynman diagrams. () diagrams are made of lines and vertices at which four lines meet; () for each vertex assign a factor of λ; (3) for each line assign /m ; (4) for each external end assign J. So the term in Z(J) corresponds to 7 lines, vertices, and 6 ends, is ( ) 7 m ( λ) J 6, which can be derived from the writing terms as derivatives trick above by taking the J 4 term from e J /m, 7, which is J 7! m and the λ term from e 4!( λ d dj ) 4 (, which is λ d ) 4. 4! dj Acting the latter on the former, we get λ 4! J 6 4! 7!6! 7 m 4. (Notice that twice the number of lines is equal to four times the number of vertices plus the number of ends.).9 Connected VS Disconnected To distinguish between connected and disconnected ones, write Z(J, λ) Z(0, λ)e W (J,λ) Z(0, λ) N0 N! W (J, λ)n. By definition, Z(0, λ) consists of those diagrams with no external source J. W is a sum of connected diagrams while Z contains connected as well as disconnected diagrams. N is the number of disconnected pieces. 5
3 Quantum Field Theory Z Dφ e iiφ. Consider Iφ d 4 x ( aφ a φ ) m φ. We add into the action a source term J(x)φ. ZJ Dφ exp i : dφ exp x,t ( φ) ) m φ + Jφ d 4 x ( ) φaφ + Jφ ( d 4 x i i (det A) / exp d 4 x JA J i : (det A) / exp d 4 x d 4 y J(x)G(x, y)j(y). A : + m is an operator hence A is a Green s function. G F is chosen to be the Feynman propagator G F (x, y) d 4 p e ip (x y) (π) 4 p + m 0 T φ(x)φ(y) 0. iɛ Elaboration / Proof : An n-point correlation function is defined as ( ) lim 0 T φ(x ) φ(x n ) exp T i T T Hdt 0 : 0 T {φ(x ) φ(x n )} 0 Dφ φ(x ) φ(x n )e isφ Dφ e isφ ( Z0 ij(x ) ) ( ij(x n ) ) ZJ J0 In the free scalar theory, ZJ ( i ZJ Z0 exp Dφ e isφ+ijφ. ) dx dy J(x)A (x, y)j(y) A (x, y) (x, y). + m 6
The -point correlation function is then ( ) ( ) ( i 0 T {φ(x)φ(y)} 0 exp du dv J(u)A (u, v)j(v)) ij(x) ij(y) J0 i exp du dv J(u)A (u, v)j(v) J(x) J(y) J0 i exp du dv ( (u y)a (u, v)j(v) + J(u)A (u, v)(v y) ) J0 J(x) i (dv exp A (y, v)j(v) + du J(u)A (u, y) ) J(x) J0 i (dv exp A (y, v)(x v) + du (x u)a (u, y) ) J0 i (dv exp A (x, v) J(v) + du J(u)A (u, x) ) (as above) i ( A (y, x) + A (x, y) ) A (x, y) the third last line comes from /J(x) applied to exp while keeping the integral term ( as above ) fixed. The terms vanish since they are being evaluated at J 0. We can express the -point correlation function diagrammatically by a line joining points x and y. Similarly, the 4-point correlation function is 0 T {φ(x)φ(y)φ(z)φ(w)} 0 ( ) ( ) ( ) ( ) ( i exp ij(x) ij(y) ij(z) ij(w) dx dy J(x)A (x, y)j(y)) () 4 ia (x, y)ia (z, w) + ia (x, z)ia (y, w) + ia (x, w)ia (y, z) exp A (x, y)a (z, w) + A (x, z)a (y, w) + A (x, w)a (y, z). ( i J0 J0 dx dy J(x)A (x, y)j(y)) We can express the 4-point correlation function diagrammatically by a line joining x and y, a line joining z and w, and similarly for the other two terms. The interacting -point correlation function is 0 T {φ(x )φ(y)} 0 Z int 0 Z int J For the φ 4 theory with potential V (φ) λ 4! φ4, ( ) exp V exp λ ij 4! ( ) ( ) ij(x) ij(y) Z int J J0 ( ) Dφ e is freev (φ)+ijφ exp V Z free J. ij du ( ) 4 i λ ij(u) 4! du ( ) 4 + ij(u) J0 7
the first equality comes from the relation G ( ) ( b F (b) F x) G(x)e x b x0. Hence, ( ) ( ) Z int J ij(x) ij(y) J0 ( ) ( ) ( i λ ( ) ) 4 du Z free J ij(x ) ij(y) 4! ij(u) ( Z free 0 Z free 0 ij(x ) ) ( ij(y) ia (x, y) iλ iλ(a (x, y)) ) ( i λ 4! du ( ij(u) ) 4 ) J0 ( i exp dz(a (x, z))(a (z, z))(a (y, z)) dz(a (z, z))(a (z, z)) dx dy J(x)A (x, y)j(y)) We can express the -point interacting correlation function diagrammatically by a line joining x and y (first term), a line with a loop attached (second term), a disconnected diagram of a line joining x and y and a vacuum diagram (diagram without external legs) in the shape of the number 8 (third term). J0 and In summary, ( ) ( ) Z int J ij(x) ij(y) connected diagrams x vacuum bubbles J0 0 T {φ(x )φ(y)} 0 connected diagrams. To go to momentum space, calculate dx dy e ipx e qx 0 T {φ(x)φ(y)} 0 dx dy e ipx e qx (A (x, y)) dx dy e ipx e qx dk (π) 4 dk dx dy (π) 4 dy dk k + m ek(x y) k + m eix(p k) e iy(k q) k (p k) eiy(k q) + m dk k + m (π)4 (p k)(k q) k + m (π)4 (p q) (π) 4 (p q) imposes the overall momentum conservation, and we use the momentum representation of the propagator k +m. 3. The Schwinger Way Z(J) Dφ e i d 4 x ( φ) m φ λ 4! φ4 +Jφ e λ 4!i d 4 w ij(w) 4 Z(0, 0) e λ 4!i d 4 w Dφ e i d 4 x ( φ) m φ +Jφ ij(w) 4 e i d 4 xd 4 yj(x)d(x y)j(y) 8
D(x y) and the next steps are more complicated than... d 4 k e ik (x y) (π) 4 k m + iɛ 3. The Wick(ed) Way Z(J) Dφ e i d 4 x ( φ) m φ λ 4! φ4 +Jφ i s Z(0, 0) dx dx s J(x ) J(x s )G (s) (x,, x s ) s! s0 i s dx dx s J(x ) J(x s ) Dφ e i d 4 x ( φ) m φ λ 4! φ4 φ(x ) φ(x s ). s! s0 G(x, y) : Dφ e i d 4 x ( φ) m φ λ 4! φ4 φ(x)φ(y). Z(0, 0) G(x, y, z, w) : Dφ e i d 4 x ( φ) m φ λ 4! φ4 φ(x)φ(y)φ(z)φ(w). Z(0, 0) Hence to first order in λ, ( iλ ) d 4 w Dφ e i d 4 x ( φ) m φ λ 4! φ4 φ(x)φ(y)φ(z)φ(w)φ(w) 4 Z(0, 0) 4! λ d 4 w D(x w)d(y w)d(z w)d(w w) D(x a w) d 4 k a e ±ika(xa w) (π) 4 ka m + iɛ. And since we have the freedom of associating with the dummy integration variable either a plus or a minus sign in the exponential, we have d 4 w e (k+k k3 k4)w (π) 4 (4) (k + k k 3 k 4 ). 3.3 Scattering Amplitudes Finally consider scattering amplitudes. q S p 4E q E p 0 aq Sa p 0 dx dy e py e iqx 0 T {φ(x)φ(x)} 0 a p 0 dx φ(x)e px 0. Hence the n-particle scattering amplitudes are simply the Fourier transforms of the n-point correlation functions. 9
3.4 Appendix: Feynman Rules. Draw a Feynman diagram of the process and put momenta on each line consistent with momentum conservation.. Associate with each internal propagator p + m iɛ ( /p + m) p + m iɛ η ab p iɛ (scalar propagator); (fermion propagator); (photon propagator). 3. Associate with each interaction vertex a coupling constant (scalar:λ, scalar-fermion:g, fermion-photon:eγ a, etc) and ( (π) 4 (4) p in ) p out. in out 4. Integrate over momenta associated with loops, with the measure and multiplied by - for fermions and for bosons. d 4 k (π) 4, 5. Add in wavefunction terms for external particles of momentum p and spin s: u(p, s) incoming fermions; ū(p, s) outgoing fermions; v(p, s) incoming antifermions; v(p, s) outgoing antifermions; ɛ a incoming photons; ɛ a outgoing photons. 6. Take into account symmetry factors, which originate from various combinatorial factors counting the different ways in which the J s can hit the J s. 0