he Quantum heory of Finite-emperature Fields: An Introduction Florian Divotgey Johann Wolfgang Goethe Universität Franfurt am Main Fachbereich Physi Institut für heoretische Physi 1..16
Outline 1 he Free Partition Function Interacting Fields 3 Infrared Divergences
he Free Partition Function
General Remars Consider isolated system Σ consisting of two disjoint subsystems Σ 1 and Σ. Assume that Σ 1 Σ and that heat and particle exchange between both systems is allowed. Σ Σ 1 Σ Figure: Isolated system Σ with subsystem Σ acting as heat and particle reservoir for Σ 1. After certain time t, both systems will have same temperature and same chemical potential µ: Σ 1 and Σ are in thermal and chemical equilibrium. Systems with fixed, V, and µ are described by grand canonical ensemble. Statistical properties of a grand canonical ensemble are encoded in the grand canonical partition function [ Z(, V, µ) = r exp 1 ( Ĥ )] µ i ˆNi, i where Ĥ ˆ= Hamilton operator of the system, ˆN i ˆ= conserved number operators of the system. Goal: Find functional integral representation of Z(, V, µ)!
Free Partition Function for Real-Valued Scalar Field heory (I) System shall be described by L (φ, µφ) = 1 [ µφ(t, r) ] µ φ(t, r) m φ (t, r) + L int(φ, µφ), where φ : R 4 R, (t, r) φ(t, r). Set L int(φ, µφ) = for the moment, such that H (π, φ) = 1 π (t, r) + 1 where π(t, r) = φ(t, r). [ φ(t, r) ] φ(t, r) + m φ (t, r), Postulate existence of the following eigenvalue equations of the Schrödinger-picture field operators ˆπ(t, r), ˆφ(t, r) ˆφ(r) φ(r) = φ(r) φ(r), ˆπ(r) π(r) = π(r) π(r). Eigenstates are considered to be orthonormal and complete, i.e., φ i(r) φ j(r) = δ [φ i(r) φ j(r)], π i(r) π j(r) = δ [π i(r) π j(r)], dπ(r) dφ(r) φ(r) φ(r) = π(r) π(r) = 1, π r r with their overlap defined as φ i(r) π j(r) = exp [ i ] d 3 r π i(r)φ j(r).
Free Partition Function for Real-Valued Scalar Field heory (II) race can then be evaluated in the basis of the field operator eigenstates [ Z(, V ) = dφ(r) φ(r) exp 1 Ĥ( ) ] ˆφ, ˆπ φ(r) r Matrix element can be interpreted as transition amplitude of an initial state φ i(r), t i prepared at time t i = into a final state φ f (r), t f prepared at t f = i/ Basis of imaginary time formalism! [ φ f (r), t f φ i(r), t i = φ f (r) Û(t f, t i) φ i(r) = φ f (r) exp iĥ( ) ] ˆφ, ˆπ (tf t i) φ i(r) [ t i =, t f = i/ = φ f (r) exp 1 Ĥ( ) ] ˆφ, ˆπ φ i(r). Important: race imposes periodic boundary conditions in time-lie direction, i.e., φ f ( i/, r) = φ i(, r). General transition amplitude φ f (r), t f φ i(r), t i can be written as functional integral φ f (r), t f φ i(r), t i φ(t f,r)=φ f (r) { tf = Dφ(t, r) Dπ(t, r) exp i t i φ(t i,r)=φ i (r) thus φ(r), i/ φ(r), φ( i/,r)=φ(r) = Dφ(t, r) φ(,r)=φ(r) dt { i/ Dπ(t, r) exp i dt [ d 3 r π(t, r) φ(t, r) H (π, φ)] }, [ d 3 r π(t, r) φ(t, r) H NKG (π, φ)] }.
Free Partition Function for Real-Valued Scalar Field heory (III) Finally, Wic-rotating to Euclidean space-time and integrating the conjugate momentum field out of the theory yields Z(, V ) { 1/ [ ] } = Dφ E (τ, r) Dπ E (τ, r) exp dτ d 3 r iπ E (τ, r),e φ E (τ, r) H E (π E, φ E ) periodic { 1/ } = N Dφ E (τ, r) exp dτ d 3 r L E (φ E, µ,e φ E ), periodic where L E (φ E, µ,e φ E ) = 1 [ µ,e φ E (τ, r) ] µ,e φ E (τ, r) + m φ E (τ, r).
Evaluation of the Functional Integral (I) Evaluate functional integral in momentum space. Fourier decomposition of real-valued scalar field is given by with φ E (n, ) = φ E ( n, ). φ E (τ, r) = 1 1 V n φ E (n, )e i(ωnτ+ r), Periodic boundary conditions lead to so-called bosonic Matsubara frequencies φ E (1/, r) =! φ E (, r) e iωnτ =! 1 ω b n ωn = π n. he exponent becomes 1 1/ { dτ d 3 r [ µ,eφ E (τ, r) ] } µ,eφ E (τ, r) + m φ E (τ, r) ( ) = 1 ω b n + E φe (n, ) φ n E (n, ). he functional integral measure becomes Dφ E (τ, r) = dφ E (τ, r) = N ( ) ] ] dre[ φe (n, ) dim[ φe (n, ), τ r n> where the Jacobian N ( ) arises from the normalization of the Fourier decomposition.
Evaluation of the Functional Integral (II) Finally, the partition function can be evaluated as ] Z(, V ) = N dre[ φe (n, ) exp n> n> n> ] dim[ φe (n, ) exp n> ( ) ω b n + E ( ) ω b n + E ( ) = N ω b 1/ ( ) n + E ω b 1/ n + E n> n> ( ) = N ω b 1/ n + E n [ ] Re φe (n, ) [ ] Im φe (n, )
Physical Observables (I) hermodynamical observables of a system can be derived from a thermodynamical potential. In case of the grand canonical ensemble, the potential of interest is given by the grand canonical potential, defined as Ω(, V, µ) = ln Z(, V, µ). he entropy S, the pressure p, and the particle number N can be determined as S = Ω, p = Ω V,µ V, N = Ω,µ µ.,v For the real-valued scalar field, the grand canonical potential may be written as ( ) ω b 1/ n + E Ω(, V ) = ln Z(, V ) = ln n ( ) = ω b n + E ln n = [ E ( )] + ln 1 e E / V d 3 [ E ( )] = V (π) 3 + ln 1 e E /.
Physical Observables (II) First term corresponds to the zero-point energy and requires renormalization. he second integral corresponds to the thermal contribution and can be solved analytically in the massless limit, m, d 3 ( ) (π) ln 1 e / = 3 = 3 6π d dω (π) 3 dξ ln (1 ) e / ξ 3 e ξ 1 = π 9 3. hen, such that Ω m=(, V ) = π 9 4 V, S m= = π 45 3 V, p m= = π 9 4, N =.
Interacting Fields: Partition Function and Perturbative Expansion
Partition Function for Interacting Fields Now, consider the case where L int(φ, µφ) = λ 4! φ4 (t, r). Similar to the case of free fields, one obtains { 1/ Z(, V ) = N Dφ E (τ, r) exp dτ d 3 r [L E (φ E, µ,e φ E ) + λ4! ]} φ4e (τ, r) periodic Same problem as in vacuum QF: Functional integral cannot be solved analytically anymore! Solution: Introduce source J E : R 4 R, (τ, r) J E (τ, r) and express interaction term through functional derivatives w.r.t. these sources. where Z(, V ) L int Z (λ) [J E,, V ), Z (λ) [J E,, V ) { 1/ = N Dφ E (τ, r) exp dτ d 3 r [L E (φ E, µ,e φ E ) + λ4! ] φ4e (τ, r) periodic 1/ } + dτ d 3 r J E (τ, r)φ E (τ, r) [ 1 1/ = N dτ d 3 r λ ( ) δ 4 ] j j! j= 4! δj E (τ, r) { 1/ 1/ } Dφ E (τ, r) exp dτ d 3 r L E (φ E, µ,e φ E ) + dτ d 3 r J E (τ, r)φ E (τ, r) periodic
Finite-emperature Feynman Rules (I) Perturbation theory is usually done in momentum space. ranslate Z[J E,, V ) into momentum space and derive momentum space Feynman rules. Fourier decomposition of source is given by J E (τ, r) = 1 1 3 3 n V with J E (n, ) = J E ( n, ). J E (n, )e i ( ω b n τ+ r ), hen, the exponent can be written as 1/ 1/ dτ d 3 r L E (φ E, µ,e φ E ) + dτ d 3 r J E (τ, r)φ E (τ, r) = V { 1 [ φ E (n, ) ( ) ] ω b φe ( n, ) n + E + J E (n, ) n V V 3 V Now, using the tric that J E (n, ) V 3 n = 1 [ JE (n, ) 3 n V φ E ( n, ) V φ E ( n, ) V + J E ( n, ) V 3 φ E (n, ) V ], } φ E ( n, ). V
Finite-emperature Feynman Rules (II) then... = V { 1 [ φ E (n, ) ( ) ] ω b φ n + E E (n, ) n V V + 1 [ JE (n, ) φ E (n, ) + J ]} E (n, ) φ E (n, ) 3 V V 3 V V [ ] = V [ ( ) ] [ ] [ ] n> ω b Re φe (n, ) Re JE (n, ) Re φe (n, ) n + E + 3 V V V [ ] + V [ ( ) ] [ ] [ ] n> ω b Im φe (n, ) Im JE (n, ) Im φe (n, ) n + E + 3 V V V = n> + V n> ( ) ω b n + E { [ [ ] } Re φe (n, )] + Im φe (n, ) J E ( n, ) () 3 E (n, ) J E (n, ), 3 V V where the Euclidean Matsubara propagator of the real-valued scalar field is defined as () E (n, ) = 1. (ωn b ) + E
Finite-emperature Feynman Rules (III) hen, the functional integral can be solved as { 1/ 1/ Dφ E (τ, r) exp dτ d 3 r L E (φ E, µ,e φ E ) + dτ periodic { 1 V J E ( n, ) = exp () 3 E (n, ) J } E (n, ) 3 n V V ( ) ] dre[ φe (n, ) exp n> ω b n + E n> ] dim[ φe (n, ) n> { 1 = Z (λ=) V [,, V ) exp n where exp n> ( ) ω b n + E J E ( n, ) () 3 E (n, ) J E (n, ) 3 V V ( ) Z (λ=) [,, V ) = ω b 1/ n + E n is simply the result of the non-interacting case. } d 3 r J E (τ, r)φ E (τ, r) [ ] Re φe (n, ) [ ] Im φe (n, ) },
Finite-emperature Feynman Rules (IV) he Dyson series can be written as [ 1 1/ dτ d 3 r λ ( ) δ 4 ] j j! j= 4! δj E (τ, r) 1 = λ φ E (n 1, 1)... φ E (n 4, 4) j! 4! j= n 1,...,n 4 1,..., V V 4 V ] j δn 1 +...+n 4,δ(3) 1 +...+ 4, 1 = λ ( ) 3 δ V j! 4! V j= n 1,...,n δ J 4 1,..., E ( n 4 1, 1) ( ) j 3 δ V... V V δ J E ( n 4, 4) δn 1 +...+n 4,δ(3) 1 +...+ 4,]. Finally, the momentum space partition function for φ 4 -theory reads Z (λ) [ J 1 E,, V ) = N λ ( ) 3 δ V j! 4! V j= n 1,...,n δ J 4 1,..., E ( n 4 1, 1) ( ) ] j 3 δ V... V V δ J E ( n 4, 4) δn 1 +...+n 4,δ(3) 1 +...+ 4, { 1 Z (λ=) V J E ( n, ) [,, V ) exp () 3 E (n, ) J } E (n, ). 3 n V V
Finite-emperature Feynman Rules (V) he finite-temperature Feynman rules in momentum space can be obtained from the leading order term in the perturbative expansion of Z (λ) [ J E,, V ) in powers of the coupling constant λ: (i) Draw all topologically inequivalent diagrams at a certain order in perturbation theory, (ii) determine the numerical and/or combinatorial factor for each diagram, (iii) multiply each vertex by a factor of λ V 4! δn f,n i δ(3) f,, representing the i conservation of the Matsubara frequencies/momenta at each vertex. () (iv) insert a Matsubara propagator, E (n, ), for each internal line, (v) sum over all internal Matsubara frequencies/momenta, V. n Using the above Feynman rules, one obtains Z (λ) [ J E,, V ) = Z (λ=) [,, V ) 1 + 3 + 6 + + O(λ )
Physical Observables to Leading Order in λ (I) As in the non-interacting case, the physical quantities to all orders in perturbation theory can be derived from the grand canonical potential Ω (λ) (, V ) = ln Z[ J E =,, V ). wo observations: (i) Due to the logarithm, only connected diagrams may contribute to physical quantities. (ii) Due to J E =, only diagrams without external legs may contribute to physical observables. o leading order in λ, the grand canonical potential is given by Ω (1) (, V ) = ln Z (λ=1) [ J E =,, V ) = ln Z () [,, V ) 1 + 3 = ln Z () [,, V ) 3
Physical Observables to Leading Order in λ (II) Using to momentum space Feynman rules, the vacuum diagram can be expressed as = λ [ V () E 4! V (n, )]. n A central technical aspect of finite-temperature field theory is the evaluation of Matsubara sums. his, in general, may become arbitrarily complicate. A powerful tool is given by the methods of complex calculus, which can be used to show that () V E (n, ) = V n d 3 1 (π) 3 E where 1 n B (E ) = e E / 1 defines the Bose-Einstein-distribution. [ 1 + n B (E ) Similar to the non-interacting case, the first term corresponds to an infinite vacuum contribution, while the thermal contribution can be solved analytically in the massless case, m d 3 n B ( ) = dω (π) 3 d 1 1 e / 1 = ξ dξ π e ξ 1 = 1. ],
Physical Observables to Leading Order in λ (III) Finally, the vacuum diagram is given by = λ ( ) V = λ 4! 1 4!144 3 V. hen, the grand canonical potential for massless φ 4 -theory to leading order in the coupling constant λ is given by ( Ω (1) m= (, V ) = 4 V π 9 + λ ), 4!48 such that the entropy and the pressure at this order are determined as ( S (1) π m= = 3 V 45 λ ) (, p (1) π m= 3!48 = 4 9 λ ). 4!48
Infrared Divergences: hermal Mass and Breadown of Naive Perturbation heory
Infrared Divergences At NLO in the coupling constant λ, the following vacuum diagram will have a contribution to physical observables [ ] = λ V () (4!) E (l, ) () () E (n, p) E ( n, p). V V l n p Observation: Inner loop completely decouples from the tadpoles. Consider massless φ 4 -theory and estimate contribution of Matsubara zero-mode d 3 ( ) p 1 = dω (π) 3 p (π) 3 dp 1 p. Diagram is IR divergent due to the factorization of the (massless) Matsubara propagators of the inner loop. Same estimation in massive φ 4 -theory d 3 ( ) p 1 = p dω (π) 3 p + m (π) 3 dp (p + m ) = π m dξ ξ (1 + ξ ) = 8πm. IR divergence is regularized by a non-vanishing mass!
hermal Mass he full Matsubara propagator of the theory is defined as the sum of all two-point functions with all possible 1PI insertions = + + +..., 1 (), 1 E (n, ) = E (n, ) Π E. Up to LO in λ, the 1PI contribution Π (1) E 1 = λ () V E (n, ) = λ V n m= = λ 4. is given by the tadpole diagram d 3 [ ] 1 1 + n (π) 3 B (E ) E Obviously, the massless scalar field generates a thermal mass, which is - up to LO - given by ( ) (1), 1 (), 1 (1) E (n, ) = E (n, ) Π E = + ω b n + m, with m = λ /4.
Breadown of Naive Perturbation heory (I) At arbitrary order λ N in perturbation theory, the dominant IR divergence is given by a vacuum diagram of the following form ( ) ( 3!)N V N ( ) [ λ N = N 4! V l n=,v () E (n, p) 1 V N [ ] N d 3 p Π(1) E (π) 3 ( ) 1 N p ] N ( () E (n, ) V Now, sum over diagrams of this particular type 1 V [ ] j d 3 p [ ] j Π(1) E () j (π) j= n 3 E (n, p) = V d 3 p 1 [ ] Π(1) () j (π) n 3 E E j (n, p) Π(1) E j=1 = V { [ ] d 3 p m ln 1 + (π) n 3 (ωn b ) + p ) N n () E m (ω b n ) + p p (n, p) } () E (n, p)
Breadown of Naive Perturbation heory (II) Finally, evaluate term containing the Matsubara zero-mode V [ ( ) ] d 3 p ln 1 + m m = V ( dω (π) n 3 p p (π) 3 dξ ξ [ln 1 + 1 ) 1 ] ξ ξ = m3 1π V λ3/. Obviously, the sum of ring diagrams contributes at O(λ 3/ ) and remains finite. Naive perturbation theory breas down due to non-analytic powers of the coupling constant λ. Resummation of IR divergences needed at finite temperature!