Problem Set 9 for Quantum Field Theory Course

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 βH i βH Z Tr e− DΨ Ψ e− Ψ , (1) = = 〈 | | 〉

with β 1/k T is the inverse temperature and the trace is performed by integrating over the continuum = B space of eigenfunctions Ψ(x).

(a) Write the above expression as a functional integral of a ﬁeld Φ(x,τ) 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) = 2m + and a indices the discretization. Use τ i t, Hamilton’s equations and the derivation from the = lecture to bring (2) to the form below: Z R β dτL (q) N Dq(τ)e− 0 E . (4) q(β) q(0) =

What is the Euclidean Lagrangian LE ?

1 β (c) Calculate S R dτL for the quantum harmonic oscillator with unit mass E = 0 E

1 ¡ 2 2 2¢ LE q˙ ω q . (5) = 2 + Use a Fourier decomposition that satisﬁes the periodic boundary conditions:

X∞ cn 2πinτ/β q(τ) p e . (6) = n β =−∞

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 + , (7) = p2

and perform the Gaussian integrals. By using the identity

µ 2 ¶ sinhx Y∞ x 1 2 2 (8) x = n 1 + n π = show that this path integral (up to a β-dependent normalization factor) coincides with the partition function of the quantum harmonic oscillator. P βω(n 1/2) Hint: from statistical physics we know it is given by Z n∞ 0 e− + . = = 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 Dφe− E , (9) =

where SE is called the Euclidean action. For a free scalar ﬁeld it reads

Z Z β d 1 ¡ 2 2 2 2¢ SE d x dτ φ˙ ( φ) m φ . (10) = 0 2 + ∇ + (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) ∼ − τ − ∇ + (b) This determinant can be calculated explicitly with the help of mode expansion. Write φ(τ,x) as

2πinτ/β Z d X∞ e d k φ(τ,x) eikxφ . (12) p d nk = n β (2π) =−∞ Write this back into (10) and use the tricks of Problem 9.1(d) to show that the result is

βω /2 Y e− k Z C , (13) = 1 e βωk k − − where C is a β-dependent constant and p ω k2 m2 . (14) k = +

2 (c) Do a similar calculation for a fermionic oscillator. The Euclidean action is

Z β ³ ´ SE dτ ΨΨ˙ ωΨΨ . (15) = 0 + If we impose anti-periodic boundary conditions, the mode expansion looks like

1 X∞ 2πi(n 1/2)τ/β Ψ p Ψne + , (16) = β n =−∞ and similarly for Ψ. Utilize these expressions to write · ¸ X∞ 2πi SE Ψn (n 1/2) ω Ψn . (17) = n β + + =−∞ (d) Calculate the partition function Z S Z DΨDΨe− E . (18) = 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 ¶ Y∞ x coshx 1 2 2 . (19) = n 0 + π (n 1/2) = + Show that in the end the partition function is

Z C(β)cosh¡βω/2¢, (20) = 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 βH ´ GF (τ1,τ2) Tr e− T (Ψ(τ1)Ψ¯ (τ2)) (21) = Z Using the relations

T (Ψ(τ )Ψ¯ (τ )) θ(τ τ )Ψ(τ )Ψ¯ (τ ) θ(τ τ )Ψ¯ (τ )Ψ(τ ), (22) 1 2 = 1 − 2 1 2 − 2 − 1 2 1 τH τH Ψ(τ) e Ψ(0)e− (23) = show that G (0,τ ) G (β,τ ) (24) F 2 = − F 2 implying Ψ(β) Ψ(0). (25) = − 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 φ4 theory. In Minkowski space, this theory is described by the Lagrangian density

1 2 1 2 2 λ 4 LM (∂µφ) m φ φ . (26) = 2 − 2 − 4! The action is given by Z S d 4xL . (27) M = M

3 (a) Perform a Wick rotation in time x4 i x0 (28) = and calculate the Euclidean action SE and Lagrangian LE Z S d 4x¯L , (29) E = E

where x¯ (x1,x2,x3,x4). = (b) Working with the free Lagrangian

1 2 1 2 2 LE0 ( φ) m φ (30) = 2 ∇ + 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 Dφφ(x¯1)φ(x¯2)e− E 0 T φ(x¯ )φ(x¯ ) 0 . (31) 1 2 R S 〈 | | 〉 = Dφe− E

1 2 1 2 2 λ 4 LE ( φ) m φ φ . (32) = 2 ∇ + 2 + 4! Expand the exponential with the interaction term up to ﬁrst order in λ 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) = − + 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 = − 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 ωk . (35) 〈 〉 = k 2ħ

Show that using the dispersion relation ω ck the energy E stored between the plates can be k = written as " # Z 2 r 2 c 2 d k X∞ 2 ³nπ´ ħ k E L 2 k 2 k , (36) = 2 (2π) | k| + n 1 k + a = 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 = (why?).

4 (b) Write the same energy in the V aL2 volume without the presence of these plates. = Z 2 Z q c 2 d k dkz 2 2 E0 ħ L k a 2 k kz . (37) = 2 (2π)2 2π k +

Perform a variable transform nπ kz (38) = 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 2π). This integral is = | k| divergent. Let us introduce a cutoff-function Λ(k) such that Λ(0) 1, Λ( ) 0, and Λ(n)(0) = ∞ = = Λ(n)( ) 0 for any n 0th derivative of Λ(k). ∞ = > By multiplying all divergent contributions with Λ bring the integral to the form " # Z r 2 Z r 2 ∆E c ∞ k X∞ ³nπ´ ∞ ³nπ´ ħ 2 2 2 kdk Λ(k) k Λ(p) dn k Λ(p) , (40) L = 2π 0 2 + n 1 + a − 0 + a = where we used the brief notation Ãr ! ³nπ´2 Λ(p) Λ k2 . (41) = + a

(d) Introduce a new variable such that a2k2 u , (42) = π2 so the energy difference per unit area is

2 · Z ¸ cπ 1 X∞ ∞ ħ 3 F (0) F (n) dn F (n) . (43) 4a 2 + n 1 − 0 = What is F (n)? (e) For taking difference of the sum and integral use the Euler–Maclaurin formula:

n Z n X f (n) f (m) X B2k h (2k 1) (2k 1) i f (j) f (x)dx + f − (n) f − (m) R , (44) j m = m + 2 + k 1 (2k)! − + = =

where Bi are Bernoulli numbers and R is the residual difference of the two sides. Keep the ﬁrst non-vanishing term (with respect to the summation over k) and exploiting the properties of Λ(k) show that ∆E cπ2 ħ . (45) L2 = −720a3 Note that the result is independent of the regulator function Λ(k). Hint: it is useful to introduce a variable transform in the integral in F (n) such that its argument appears only once in the expression. (f) Calculate the force over unit area F ∂(∆E/L2) (46) L2 = − ∂a to show that the plates experience an attractive force.

5 1 g = =

Figure 1: Feynman rules in zero space-time dimensions.

Problem 9.5 Convergence of perturbative series Perturbation theory is a very efﬁcient method to obtain powerful predictions from Quantum Field Theory. It is interesting to ask some fundamental questions about the convergence of these perturbative series. We are going to investigate this topic with the help of a zero-dimensional model. Let us deﬁne a φ4-like theory with the help of the diagrammatic rules shown in Fig. 1. The m-point function can be expanded as

X∞ p (p) Gm g Gm , (47) = p 0 = (p) so Gm is the number of diagrams with m external legs and p vertices since propagators are 1.

(a) We can write the generating functional as

m m X∞ i j Z [j] Gm , (48) = m 0 m! = where i is the complex unit. Show by counting the number of graphs Gfree G(0) in the free g 0 m = m = case that Z [j] exp£ j 2/2¤. (49) free = − (b) By performing a Fourier transform one can write the generating functional of the interacting theory (applying formulae of Problem 8.5 to the zero-dimensional case)

1 Z µ a2 g a4 ¶ Z [j] ∞ da exp i a j . (50) = p2π − 2 − 4! + −∞ Use this to write the 2m-point functions as Z 1 ∞ 2m S(a)/ G2m daa e− ħ , (51) = p2π −∞ where we reintroduced and ħ 1 g S(a) a2 a4 (52) = 2 + 4! is the zero-dimensional action. Rescale the variable a b 1/2 to show that we get a g coefﬁcient = ħ ħ for the interaction term. This means that perturbative expansion in g is equivalent to an expansion in . ħ (c) We are going to illustrate limitations of perturbative expansion in on ħ Z 1 ∞ S(a)/ Z [0] dae− ħ , (53) = p2π −∞ since this can be easily modiﬁed for the case of Green’s functions. Perform a variable transform to u(a) such that p u(a) S(a), if a 0 (54) = p ≥ u(a) S(a), if a 0. = − < Write Z [0] as an integral over u.

6 (d) In the integral 1/u0 appears. Expand it as a power series in u

1 X∞ r (u0)− cr u . (55) = r 0 =

It has a ﬁnite convergence radius rC on the complex u plane. Perform the integration over u order by order to get p X∞ p Z [0] ħ c2p Γ(p 1/2). (56) = 2 p 0 ħ + = Use the deﬁnition of the convergence radius

1/2p lim c2p rC (57) p →∞ | | = to show that expression (56) is divergent for any 0. ħ 6= Remark: This is a consequence of integrating beyond the convergence radius in u, thus the interchange between integral and summation leads to divergence. In an analogous way this leads to divergence of perturbative expansions in non-zero dimensional QFT as well. That, however involves some additional technicalities, the interested reader is referred to Section 5.5 of Rivers’ book and to [1].

References

[1] Zinn-Justin, J. Perturbation series at large orders in quantum mechanics and ﬁeld theories: Application to the problem of resummation. (1981) Phys. Rep. 70C 109 No 2. https://doi.org/10.1016/0370-1573(81)90016-8.