Problem set 9 for Quantum Field Theory course 2019.04.16. Topics covered • Path integrals at finite temperature • Fermionic and bosonic path integrals • Euclidean path integral and Feynman rules • Casimir effect • Convergence of perturbative series Recommended reading • Peskin–Schroeder: An introduction to quantum field theory Section 9.5 • R.J. Rivers: Path integral methods in quantum field theory Sections 1.8, 5.3 Problem 9.1 Finite temperature path integral: harmonic oscillator Systems at finite temperature are described by the quantum statistical partition function: Z h ¯H i ¯H Z Tr e¡ Dª ª e¡ ª , (1) Æ Æ h j j i with ¯ 1/k T is the inverse temperature and the trace is performed by integrating over the continuum Æ B space of eigenfunctions ª(x). (a) Write the above expression as a functional integral of a field ©(x,¿) by observing that the inverse temperature can be thought of as imaginary time. What are the boundary conditions? (b) This procedure of working with imaginary time is the Wick rotation at the level of path integrals. It is instructive to perform this rotation on a simple problem: Z R t f P i dt a [q˙(a)p(a) H(p(a),q(a))] DqDpe ti ¡ , (2) where p2 H(p,q) V (q), (3) Æ 2m Å and a indices the discretization. Use ¿ i t, Hamilton’s equations and the derivation from the Æ lecture to bring (2) to the form below: Z R ¯ d¿L (q) N Dq(¿)e¡ 0 E . (4) q(¯) q(0) Æ What is the Euclidean Lagrangian LE ? 1 ¯ (c) Calculate S R d¿L for the quantum harmonic oscillator with unit mass E Æ 0 E 1 ¡ 2 2 2¢ LE q˙ ! q . (5) Æ 2 Å Use a Fourier decomposition that satisfies the periodic boundary conditions: X1 cn 2¼in¿/¯ q(¿) p e . (6) Æ n ¯ Æ¡1 Hint: q is real so there is a relation between the complex cn coefficients. (d) Write the result in terms of the real parameters an and bn: an ibn cn Å , (7) Æ p2 and perform the Gaussian integrals. By using the identity µ 2 ¶ sinhx Y1 x 1 2 2 (8) x Æ n 1 Å n ¼ Æ show that this path integral (up to a ¯-dependent normalization factor) coincides with the partition function of the quantum harmonic oscillator. P ¯!(n 1/2) Hint: from statistical physics we know it is given by Z n1 0 e¡ Å . Æ Æ Problem 9.2 Finite temperature path integral: free boson and fermion In the previous exercise we have seen that we can use the path integral formalism for finite temperature systems by performing a Wick rotation. We get for the partition function Z S Z DÁe¡ E , (9) Æ where SE is called the Euclidean action. For a free scalar field it reads Z Z ¯ d 1 ¡ 2 2 2 2¢ SE d x d¿ Á˙ ( Á) m Á . (10) Æ 0 2 Å r Å (a) Perform two partial integrations and use arguments from Problem 8.4 to show that the partition function is proportional to £ ¡ 2 2 2¢¤ 1/2 Z det @ m ¡ . (11) » ¡ ¿ ¡ r Å (b) This determinant can be calculated explicitly with the help of mode expansion. Write Á(¿,x) as 2¼in¿/¯ Z d X1 e d k Á(¿,x) eikxÁ . (12) p d nk Æ n ¯ (2¼) Æ¡1 Write this back into (10) and use the tricks of Problem 9.1(d) to show that the result is ¯! /2 Y e¡ k Z C , (13) Æ 1 e ¯!k k ¡ ¡ where C is a ¯-dependent constant and p ! k2 m2 . (14) k Æ Å 2 (c) Do a similar calculation for a fermionic oscillator. The Euclidean action is Z ¯ ³ ´ SE d¿ ªª˙ !ªª . (15) Æ 0 Å If we impose anti-periodic boundary conditions, the mode expansion looks like 1 X1 2¼i(n 1/2)¿/¯ ª p ªne Å , (16) Æ ¯ n Æ¡1 and similarly for ª. Utilize these expressions to write · ¸ X1 2¼i SE ªn (n 1/2) ! ªn . (17) Æ n ¯ Å Å Æ¡1 (d) Calculate the partition function Z S Z DªDªe¡ E . (18) Æ Bear in mind that these are fermionic fields so they behave as Grassmann numbers which simplifies the exponential term. A useful identity here is µ 2 ¶ Y1 x coshx 1 2 2 . (19) Æ n 0 Å ¼ (n 1/2) Æ Å Show that in the end the partition function is Z C(¯)cosh¡¯!/2¢, (20) Æ that is the partition function for a two-state system, the correct result for the case of Fermi–Dirac statistics with a single degree of freedom. (e) Bonus question: why did we have to impose antiperiodic boundary condition? Consider the Euclidean two-point function: 1 ³ ¯H ´ GF (¿1,¿2) Tr e¡ T (ª(¿1)ª¯ (¿2)) (21) Æ Z Using the relations T (ª(¿ )ª¯ (¿ )) θ(¿ ¿ )ª(¿ )ª¯ (¿ ) θ(¿ ¿ )ª¯ (¿ )ª(¿ ), (22) 1 2 Æ 1 ¡ 2 1 2 ¡ 2 ¡ 1 2 1 ¿H ¿H ª(¿) e ª(0)e¡ (23) Æ show that G (0,¿ ) G (¯,¿ ) (24) F 2 Æ¡ F 2 implying ª(¯) ª(0). (25) Æ¡ Show that the same argument leads to periodicity for bosonic fields! Food for thought: why is the antiperiodic boundary condition of the fermion field consistent with the periodic nature of the trace operation? Problem 9.3 Euclidean path integral: Feynman rules In this exercise we consider the Wick-rotated (Euclidean) path integral formulation of the Á4 theory. In Minkowski space, this theory is described by the Lagrangian density 1 2 1 2 2 ¸ 4 LM (@¹Á) m Á Á . (26) Æ 2 ¡ 2 ¡ 4! The action is given by Z S d 4xL . (27) M Æ M 3 (a) Perform a Wick rotation in time x4 i x0 (28) Æ and calculate the Euclidean action SE and Lagrangian LE Z S d 4x¯L , (29) E Æ E where x¯ (x1,x2,x3,x4). Æ (b) Working with the free Lagrangian 1 2 1 2 2 LE0 ( Á) m Á (30) Æ 2 r Å 2 derive the Feynman rule for the propagator. Hint: a similar calculation was performed at the lecture in Minkowski space-time. It can be done by calculating the functional integral R S DÁÁ(x¯1)Á(x¯2)e¡ E 0 T Á(x¯ )Á(x¯ ) 0 . (31) 1 2 R S h j j i Æ DÁe¡ E (c) Now treat the interaction term perturbatively with ¸ small enough: 1 2 1 2 2 ¸ 4 LE ( Á) m Á Á . (32) Æ 2 r Å 2 Å 4! Expand the exponential with the interaction term up to first order in ¸ and read off the Feynman rule for the vertex. (d) For an arbitrary connected graph the number of loops equals the number of vertices minus internal propagators plus 1: L P V 1. (33) Æ ¡ Å Combine this knowledge with your previous result to show that for any connected n-point function W the Minkowski and Euclidean results are simply related: W (p ,p ,...,p ) i( i)nW (p¯ ,p¯ ,...,p¯ ), (34) M 1 2 n Æ ¡ E 1 2 n where p¯ are Euclidean momenta. Problem 9.4 Casimir effect This exercise aims to demonstrate the richness of quantum field theory vacuum. The attractive force in vacuum experienced by two metallic plates is called the Casimir effect. Let us consider two metallic plates that are squares of size L, their distance (say, along the z axis) is a. (a) The ground state energy of a Hamiltonian describing an infinite set of harmonic oscillators is X 1 H !k . (35) h i Æ k 2ħ Show that using the dispersion relation ! ck the energy E stored between the plates can be k Æ written as " # Z 2 r 2 c 2 d k X1 2 ³n¼´ ħ k E L 2 k 2 k , (36) Æ 2 (2¼) j kj Å n 1 k Å a Æ where k is the wavenumber component parallel to the plates. k Hint: recall k is quantized and also that for k 0 there is only one polarization instead of two z z Æ (why?). 4 (b) Write the same energy in the V aL2 volume without the presence of these plates. Æ Z 2 Z q c 2 d k dkz 2 2 E0 ħ L k a 2 k kz . (37) Æ 2 (2¼)2 2¼ k Å Perform a variable transform n¼ kz (38) Æ a in the kz integral to get an expression involving an integral over n (similarly to sum over n in the previous expression). (c) Let us write the energy difference due to the plates over unit area as E E0 ¡ . (39) L2 Write it as an integral over k k (the angular integral gives a factor of 2¼). This integral is Æ j kj divergent. Let us introduce a cutoff-function ¤(k) such that ¤(0) 1, ¤( ) 0, and ¤(n)(0) Æ 1 Æ Æ ¤(n)( ) 0 for any n 0th derivative of ¤(k). 1 Æ È By multiplying all divergent contributions with ¤ bring the integral to the form " # Z r 2 Z r 2 ¢E c 1 k X1 ³n¼´ 1 ³n¼´ ħ 2 2 2 kdk ¤(k) k ¤(p) dn k ¤(p) , (40) L Æ 2¼ 0 2 Å n 1 Å a ¡ 0 Å a Æ where we used the brief notation Ãr ! ³n¼´2 ¤(p) ¤ k2 . (41) Æ Å a (d) Introduce a new variable such that a2k2 u , (42) Æ ¼2 so the energy difference per unit area is 2 · Z ¸ c¼ 1 X1 1 ħ 3 F (0) F (n) dn F (n) .