Warming Up to Finite-Temperature Field Theory

Transcription

1 Warming Up to Finite-Temperature Field Theory Michael Shamma UC Santa Cruz March 2016

2 Overview Motivations Quantum statistical mechanics (quick!) Path integral representation of partition function in quantum mechanics Quantum harmonic oscillator Path integral representation of partition function in quantum field theory Partition function for free scalar field and its free energy density Partition function for interacting scalar fields and free energy density

3 Motivations QFT formulated at zero-temperature What are the effects of heating up QFTs? Initial motivations found in condensed matter physics Later applications to quantum chromodynamics, cosmology, and astrophysics Quark-gluon plasmas (QGPs), heavy-ion collisions, baryogenesis, inflation and phase transitions in the early universe, stellar emission of neutrinos, etc. Advantages over kinetic/many-body theory Path integral Non-Abelian gauge interactions Lorentz invariant

4 Quantum Statistical Mechanics For quantum system described by Hamiltonian H and set of conserved number operators ˆN i that commute with H, the density matrix: ˆρ = exp[ β(h µ i ˆNi )] (natural units used throughout h = c = k B = 1) Can calculate expectation values of observables by T r[âˆρ]  = T r[ˆρ] The grand canonical partition function can be used to calculate thermodynamic properties of a system: Z = T r[ˆρ] Pressure, entropy, particle number and free energy: P = (T ln Z) (T ln Z) N i = µ i, S = T, F = T ln Z (T ln Z) V,

5 Partition Function: Harmonic Oscillator For the (bosonic) quantum harmonic oscillator, the number operator is ˆN = a a and the Hamiltonian is H = ω( ˆN ) The partition function (with µ = 0) can be written as Z = exp[ βω 2 ]T r[exp[ β(ω) ˆN]] = exp[ βω 2 ] n exp[ β(ω) ˆN] n n=0 = exp[ βω 2 ] exp[ β(ω)n] = exp[ βω 2 n=0 = [2 sinh( ωβ 2 )] 1 ](1 exp[ βω]) 1

6 Path Integral for Partition Function in QM There are few systems for which the partition function can be calculated exactly. A more useful representation is that of the path integral Start here, translate results to QFTs The partition function can be written as Z = T r[exp( βh)] = dx x exp( βh) Defining β = ɛn the exponential is split into a product of N pieces Z = dx x exp( ɛh) exp( ɛh) x Inserting 1 = dp i 2π p i p i on the left of each exponential and 1 = dx x i x i on the right (i increasing from right to left in both cases with i = 1,..., N)

7 Path Integral for Partition Function in QM the partition function becomes dxdxi dp i Z = x p N p N exp( ɛh) x N x N p N 1 2π p N 1 exp( ɛh) x N 1 x N 1 p N 2 x 2 p 1 p 1 exp( ɛh) x 1 x 1 x (1) On the very left of equation (1) above with x playing role of x i+1 we have: x i+1 p i p i exp( ɛh) x i = exp(ip i x i+1 ) p i exp( ɛh) x i = exp(ip i x i+1 ) exp(ip i x i ) exp[ ɛ( p2 i 2m + V (x i) + O(ɛ))] = exp[ ɛ( p2 i 2m ip x i+1 x i i + V (x i ) + O(ɛ))] ɛ Using the delta function at the very right of equation (1), the integral can be carried out over x yielding the partition function

8 Path Integral for Partition Function in QM Z = lim N [ N dx i dp ] [ i N O(ɛ))] exp ɛ( p2 j 2π 2m ip x j+1 x j j + V (x j ) + ɛ i=1 j=1 xn+1 =x 1 (2 The momentum integral is Gaussian and can be easily evaluated to be dp i 2π exp[ ɛ( p2 i 2m ip x i+1 x i i )] ɛ = ( m 2πɛ ) exp( m(x i+1 x i ) 2 ) (3) 2ɛ

9 Path Integral for Partition Function in QM Thus the partition function becomes Z = [ dx ] [ i lim exp N 2πɛ i=1 m j=1 ))] ɛ( m 2 (x j+1 x j ) 2 + V (x j ɛ x N+1 =x 1 In the continuum limit, defining ( m 2πɛ )N/2 = C, where C is infinite, the above equation becomes [ Z = C D[x] exp x(β)=x(0) β Compare to zero-temperature QM counterpart: Z 0 = C D[x] exp(i dtl 0 ) 0 ( m ] 2 (dx dτ )2 + V (x)) Finite-temp partition function can be achieved in four steps:

10 Wick rotation using τ = it Introduce finite-temperature Lagrangian: L = L 0 (τ = it) = m 2 ( dx dτ )2 + V (x) Restrict integration by dt β 0 dτ Impose periodicity on x(τ) by x(β) = x(0) This process is the imaginary time formalism of doing field theory Can be used also in quantum field theory and can be used for fermions and bosons

11 Path Integral for QHO To go to momentum space, write x(τ) as Fourier series: x(τ) = T n= x n exp(iω n τ) Periodicity requires e iωnβ = 1 ω n = 2πnT Impose reality on the x(τ) x (τ) = x(τ) making the x n = x n and can now write x n = a n + ib n x n = a n ib n = x n = a n + ib n a n = a n, b n = b n Letting b 0 = 0 and putting this all together, [ x(τ) = T a 0 + ] [(a n + ib n )e iωnτ + (a n ib n )e iωnτ ] n=1 (4) Fourier representation allows quadratics of the form β 0 dτx(τ)y(τ) = T 2 m,n β x n y m dτe i(ωn+ωm ) 0

12 = x n y m δ n, m = m,n n x n y n Next, using the four steps above for the action of the quantum harmonic oscillator: β 0 = mt 2 dτ m 2 [dx dx dτ dτ + ω2 x(τ)x(τ)] x n ( ω n ω n + ω 2 )x n n= and using ω n = ω n the action becomes: S = mt 2 ω2 a 2 0 mt (ωn 2 + ω 2 )(a 2 + b 2 ) n=1 Turning our attention to the product of integrals D[x] this can be rewritten by changing variables to a n, b n to obtain D[x] = det( δx(τ) ) da 0 [ da n db n ] δx n n 1

13 Defining a new constant C = C det( δx(τ) δx n ) the harmonic oscillator partition function is [ ] Z = C da 0 da n db n exp n 1 [ mt 2 ω2 a 2 0 mt ] (ωn 2 + ω 2 )(a 2 n + b 2 n) n 1 (5) Solving the Gaussian integrals above (and solving for C ) yields Remarks Z = 1 ωβ n=1 ω 2 n ω 2 n + ω 2 The partition function as well as other extensive quantities like volume, temperature, and pressure are all finite, but with the path integral, this is not immediately clear.

14 In quantum field theory, however, these divergences may remain. When calculating most physically relevant variables, this infinite constant falls out.

15 Free Scalar Fields For free scalar fields, we can follow essentially the same procedure taking x(τ) φ(τ, x) Why can we do this? The Lagrangian for a scalar field, given by L = 1 2 µφ µ φ V (φ) = ( t φ) ( iφ)( i φ) V (φ) which is a collection of near-independent harmonic oscillators (with m=1). Interact through spatial derivatives, approximately given i φ φ(t, x + ɛ e i) φ(t, x) ɛ and since the Hamiltonian depends only on terms quadratic in canonical momentum, this coupling cannot change the previous derivation fundamentally

16 Taking derivation from previous section Z = CD[φ] exp[ where φ(β, x)=φ(0, x) L = L(t = iτ) = 1 2 ( φ τ )2 + x 3 i=1 β 0 dτ d 3 L] 1 2 ( φ ) 2 + V (φ) x i As with standard QM case, expand the fields in a Fourier series: φ(τ, x) = T n= φ(ω n, x)e iωnτ In this case, it is convenient to let the directions of the spatial coordinates be momentarily finite. Denoting these directions as L i and using f(x i ) = 1 L i n i= f(n i )e ikixi

17 where k i = 2πni L i Taking the volume to infinity gives 1 = 1 L i 2π n i k i n i dki 2π where k i = 2π L i and φ(τ, x) = T V φ(τ, k) exp(iωn τ i k x) ω n k For brevity, will quote result (follow similar steps as previous section) β Z = exp( dτ L) = 0 k [ exp [ T 2V (ωn 2 + k 2 + m 2 ) φ(ω n, ] k ) 2 ] ω n (6)

18 Looks like product of harmonic oscillator partition functions Comparing to the quantum harmonic oscillator ω 2 k 2 + m 2 = E 2 k, m 1 V, x n 2 φ(ω n, k) 2 Continuing with the analogies to QHO, Z = k [ T [ = exp ln(t )+ 1 2 k n= (ω 2 n + E 2 k) 1/2 n 0 ln(ωn) n 0 n (ω 2 n) 1/2] ] ln(ωn 2 + Ek) 2 = exp( F T ) In order to obtain the free energy density, take limit at volume goes to infinity to find F lim V V = d 3 k [ (2π) 3 T 1 2 ln(ω2 n+ek) T 2 1 ] 2 ln(ω2 n) T ln(t ) ω n ω n 0

19 Interacting Theory As in zero temperature QFT, free theory is only exactly solvable theory But because of the way the thermal theory is built, we can use perturbative techniques in much the same way to approximate interactions In much the same way as was done in this course, we can write the partition function as Z = D[φ]exp[ S 0 + S I ] where S 0 is the action of the free scalar field theory and S I is the action of the interacting theory This can be written in the form Z = C D[φ]e S0 [1 S I + 1 2! S2 I 1 3! S3 I ]

20 and using S n I = D[φ]S n I e S0 D[φ]e S 0 Z = Z 0 (1 S I + 1 2! S2 I + ) Taking the logarithm, it becomes obvious that the first term in the above equation corresponds to a term proportional to the free energy density and the terms following correspond to the corrections due to interactions

21 Feynman Rules for φ 4 -theory Taking the logarithm of the the partition function for interacting fields ln Z = ln Z 0 + ln(1 S I + 1 2! S2 I + ) The second logarithm corresponds to the interacting fields, ln Z I = ln(1 S I + 1 2! S2 I + ) and can be expanded using ln(1 x) x x2 2 x3 3 to find the first order correction: ln Z 1 S I 0 = λ [ β 0 dτ d 3 x D[φ] exp( S 0 )φ 4 ] 4 D[φ] exp( S0 ) Skipping the steps of evaluating the above formula, which involve Fourier expanding the fields, the first order correction for a φ 4 - theory is ln Z 1 = 3 4 λβv [ T n d 3 p (2π) 3 1 ω 2 n + k 2 + m 2 ] 2

22 As in zero-temperature QFT, this has a diagrammatic representation Finite-Temperature Feynman Rules for φ 4 -theory Draw all connected diagrams Determine the combinatoric factor for each diagram Include a factor of T n [ d 3 p (2π) ]D 3 0 (ω n, p) Include a factor of λ for each vertex Include a factor of (2π) 3 δ( p in p out )βδ ωin,ω out for each vertex, corresponding to energy-momentum conservation. There will be one factor of β(2π) 3 δ 0 = βv left over.

23 Summary/Next Steps Calculated the partition function of the QHO using a path integral Developed the general tools necessary for calculating observable quantities Were able to extend this to a simple scalar QFT Somethings missing: chemical potential, physical behavior discussions Extend to more concepts in QFTs: gauge theories, symmetry breaking, lattice-qcd/qgp calculations...

24 References [1] Altherr, T. Introduction to Thermal Field Theory (1993). arxiv:hepph/ v1 [2] Laine, M.;Vourinen, A. Basics of Thermal Field Theory: A tutorial on perturbative calculations (2016). Springer International Publishing. ISBN [3] Kapusta, J.; Gale, C. Finite-Temperature Field Theory Principles and Applications (2006). Cambridge Monographs on Mathematical Physics, Cambridge University Press. ISBN: [4] Bellac, M.L. Thermal Field Theory (2000). Cambridge Monographs on Mathematical Physics, Cambridge University Press. ISBN: [5] Zinn-Justin, J. Quantum Field Theory at Finite Temperature: An Introduction (2000). arxiv:hep-ph/ v1

25 [6] Yang, Yuhao. An Introduction to Thermal Field Theory. Imperial College London, (2011). Thesis. Accessed online: