Euclidean path integral formalism: from quantum mechanics to quantum field theory
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1 : from quantum mechanics to quantum field theory Dr. Philippe de Forcrand Tutor: Dr. Marco Panero ETH Zürich 30th March, 2009
2 Introduction Real time Euclidean time Vacuum s expectation values Euclidean space-time Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Bosonic field theory Interacting field
3 Introduction Real time Euclidean time Vacuum s expectation values Euclidean space-time Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Bosonic field theory Interacting field
4 Classical mechanics Lagrangian mechanics. Principle of least action: δs[l] = 0. Hamiltonian mechanics. Hamilton s canonical equations q = H(q, p), p ṗ = H(q, p). q
5 Canonical quantization Classical variables, Poisson brackets: {q i, p j } = δ ij Quantum operators, Lie brackets: [X, P] = i Away from the principle of least action. Feynman: Path integrals founded on the principle of least action, but in this case the action of each possible trajectory plays a crucial role.
6 Introduction Real time Euclidean time Vacuum s expectation values Euclidean space-time Real time Euclidean time Vacuum s expectation values Euclidean space-time Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Bosonic field theory Interacting field
7 Real time Introduction Real time Euclidean time Vacuum s expectation values Euclidean space-time Pictures: Schrödinger Heisenberg Interaction Time evolution, Schrödinger equation ψ(x, t) i = Hψ(x, t) t ψ(t) = e ih(t t 0) ψ(t 0 ) Probability amplitude for a particle to move from y to x within time interval t: x e iht y
8 Real time Euclidean time Vacuum s expectation values Euclidean space-time Free particle, H H 0 = p2 2m x e ih 0t y = ( m ) 1 2 exp (i m (x y)2) 2πit 2t Particle in a potential, H = H 0 + V(x), define: x W ǫ y = U ǫ exp( ihǫ) = W ǫ, ( m ) 1 2 exp (i m 2πiǫ 2ǫ (x y)2 i ǫ ) 2 (V(x) + V(y)) W ǫ as the approximated time evolution operator if ǫ = t N is small, because exp( i (H 0 + V)t) = lim N W ǫ N.
9 Real time Euclidean time Vacuum s expectation values Euclidean space-time Inserting N 1 complete sets of position eigenstates: x e iht y = lim dx 1 dx N 1 x W ǫ x 1 x N 1 W ǫ y. N
10 Real time Euclidean time Vacuum s expectation values Euclidean space-time Rewrite as x e iht y = Dxe isǫ, where and ( m ) N 2 Dx = lim dx 1 dx N 1 N 2πiǫ S ǫ = m ((x x 1 ) (x N 1 y) 2) ( 2ǫ 1 ǫ 2 V (x) + V (x 1) V (x N 1 ) + 1 ) 2 V (y)
11 Real time Euclidean time Vacuum s expectation values Euclidean space-time S ǫ S, the action: S = t 0 ( dt m ) 2 ẋ2 V (x) Quantum mechanics amplitude as integral over all paths weighted by e is. Quantum mechanical operators infinite-dimensional integral. Green functions: G(q, t ; q, t) = dq G(q, t ; q, t )G(q, t ; q, t)
12 Euclidean time Real time Euclidean time Vacuum s expectation values Euclidean space-time Goal: Substitute e is with a real, non oscillating and non-negative function. Define imaginary (or Euclidean) time τ as t = iτ, τ > 0. Probability amplitude: x e Hτ y = Dxe S E, where S E = is the Euclidean action. t 0 ( dτ m ) 2 ẋ2 + V (x)
13 Real time Euclidean time Vacuum s expectation values Euclidean space-time Every path is weighted by e S E. Action S = and Euclidean action S E = t 0 t 0 ( dt m ) 2 ẋ2 V (x) ( dτ m ) 2 ẋ2 + V (x) are related by S t= iτ = is E, substitution d dt = i d dτ and dt = idτ.
14 Real time Euclidean time Vacuum s expectation values Euclidean space-time Classical mechanics: The path is given by δs E = 0 Quantum mechanics: All paths are possible. The paths which minimize S E, this means near δs E = 0, are the most likely. Physical units: exp( S E / ) Classical limit: 0 least action.
15 Vacuum s expectation values Real time Euclidean time Vacuum s expectation values Euclidean space-time Goal: Elegant expression for 0 A 0. Let n be the energy eigenstates, then Tr ( e Hτ A ) = e Enτ n A n n=0 and Z (τ) = Tr ( e Hτ ) = n=0 e Enτ For τ the term E 0 dominates the sums, this means Tr ( e Hτ A ) 0 A 0 = lim τ Z (τ) Mean in a canonical statistical ensemble.
16 Real time Euclidean time Vacuum s expectation values Euclidean space-time Example: Correlation functions x(t 1 ) x(t n ) 0 x(t 1 ) x(t n ) 0 Continued analytically to Euclidean times τ k = it k 1 x(t 1 ) x(t n ) = lim τ Z (τ) Dx x (τ 1 ) x (τ n )exp( S E [x (τ)]) where Z (τ) = Dx exp( S E [x (τ)]).
17 Real time Euclidean time Vacuum s expectation values Euclidean space-time Euclidean space-time Goal: Show that, using the imaginary time, we move from a Minkowski to a Euclidean space. We define the time coordinate x 4 as x 0 = ix 4, x 4 R. ( x 0, x 1, x 2, x 3) Minkowski (signature ( 1, 1, 1, 1)) ( x 1, x 2, x 3, x 4) Euclidean, because g ( x 4, x 4) = 1.
18 Introduction Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Real time Euclidean time Vacuum s expectation values Euclidean space-time Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Bosonic field theory Interacting field
19 Why quantum field theory? Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Quantum mechanics and special relativity problems Negative energy states (E 2 = p 2 + m 2 ) Assume existence of antiparticles. Quantum field theory: Wave functions replace by field operators. Field operators can create and destroy an infinite number of particles QFT can deal with many particle system.
20 Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Wightman and Schwinger functions Define the Wightman function as the n-point correlation functions: W (x 1,...,x n ) = 0 φ(x 1 ) φ(x n ) 0 Wightman functions can be continued analytically into a region of the complex plane Define the Schwinger functions as S (...; x k, x 4 k ;...) W (...; ix 4 k, x k;... ), for x 4 1 > x4 2 >... > x4 n.
21 Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Schwinger functions in Euclidean space Wightman functions in Minkowski space. W (x 1,...,x n ) = lim S (...; x k, ixk 0 + ǫ k;... ) ǫ k 0,ǫ k ǫ k+1 >0
22 Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Advantages: Schwinger functions obey to simpler properties. From symmetry property we build the representation in terms of functional integrals. ()
23 Time-ordered Green functions Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation We define the Time-ordered Green functions as τ (x 1,...,x n ) = 0 Tφ(x 1 ) φ(x n ) 0, where T is the time ordering operator. For the 2-point function we have { W (x), x 0 > 0 τ (x) τ (x, 0) = W ( x),x 0 < 0. W (x) is analytic in the lower half complex plane. W ( x) is analytic in the upper half complex plane.
24 Wick rotation Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Counter-clockwise rotation Generalization: τ (x 1,...,x n ) = lim S φ π 2 (...; x k, e iφ x 0 k ;... )
25 Introduction Bosonic field theory Interacting field Real time Euclidean time Vacuum s expectation values Euclidean space-time Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Bosonic field theory Interacting field
26 Bosonic field theory Interacting field Symmetry of Euclidean correlation functions Euclidean fields commute Like classical fields. Random variables, not operators Expectation values: F [φ] = For Euclidean functional integral: dµf [φ] dµ = 1 Z e S[φ] x dφ(x) Combining, like the.
27 Bosonic field theory Bosonic field theory Interacting field Goal: Find explicitly the expression for the Euclidean correlation φ(x 1 ) φ(x n ) in Euclidean functional integral formalism. From the Gaussian integrals we find Z 0 (J) 1 d k φexp ( 12 ) ( 1 Z (φ,aφ) + (J, φ) ( = exp J, A 1 J )) 0 2 where J is an arbitrary vector and Z 0 d k φexp ( 12 ) (φ,aφ).
28 Bosonic field theory Interacting field For a free field we have Z [J] Z 0 [J] = exp ( 1 2 ) d 4 xd 4 yj (x) G (x, y) J (y), where G(x, y) is the propagator that satisfies ( + m 2 ) G (x, y) = δ (x y). The correlation functions are given by δ n Z [J] φ(x 1 ) φ(x 2n ) = δj (x 1 ) δj (x n ). J=0
29 Bosonic field theory Interacting field We move from a discrete to a continuous representation, this means φi φ(x), J i δ δj(x). Comparing ( 1 Z 0 [J] = exp 2 ( 1 Z 0 [J] = exp 2 ) (J, GJ), ( J, A 1 J )), we find A G 1 = + m (φ,aφ) 1 ( φ,g 1 φ ) = 1 ( ( φ, + m 2 ) φ ) = S 0 [φ]. 2 2
30 Bosonic field theory Interacting field Inserting in the Gaussian integrals we find Z 0 [J] = 1 Z 0 x dφ(x) exp( S 0 + (J, φ)), where ( Z 0 = dφ(x) exp 1 ( ( φ, + m 2 ) φ )). 2 x Infinite-dimensional integral not well defined. Define the measure dµ 0 (φ) = 1 Z 0 D [φ]e S 0(φ), where D [φ] x dφ(x).
31 Bosonic field theory Interacting field Using this functional approach we find the correlation functions φ(x 1 ) φ(x n ) = dµ 0 (φ)φ(x 1 ) φ(x n ) = 1 dφ(x)e S0[φ] φ(x 1 ) φ(x n ). Z 0 x
32 Interacting field Bosonic field theory Interacting field We neglect the problems associated with divergences and renormalization. Euclidean action S [φ] = S 0 [φ] + S I [φ], where S I describes the interaction part. From Gaussian functional integral we find Z [J] = 1 dφ in (x)e S[φ]+(J,φ) Z where and φ in is the free field. x Z = dφ in (x)e S[φ], x
33 Introduction Real time Euclidean time Vacuum s expectation values Euclidean space-time Why quantum field theory? Wightman and Schwinger functions Time-ordered Green functions Wick rotation Bosonic field theory Interacting field
34 Goal: Deduce the rules of perturbation theory from functional integral. (Feynman rules) Consider a scalar field with a quartic self-interaction given by S I [φ] = g d 4 x φ(x) 4. 4! We start to expand e S I in the functional expression for an interacting field.
35 We find the functional integral Z [J] = Z [ 0 (S Z exp I δ δj (x) ]) ( ) 1 exp 2 (J, G 0J). For the associated correlations functions we find G (x 1,...,x n ) = Z [ 0 δ Z δj (x 1 )... δ (S δj (x n ) exp I exp δ δj (x) ]) ( 1 0J)) 2 J=0 (J, G.
36 Graphical interpretation of the last equation: [ ] Expand the exponential of S δ I δj(x) δ δj(x i ) Each derivative is indicated by an external point from which emerges a line ( 4 Each factor g d 4 δ x δj(x i )) is indicated by an internal vertex, from which emerges four lines. Expand exp ( 1 2 (J, G 0J) ) Internal lines
37 We can normalize the generating functional to Z (0) = 1, using [ ]) ( Z δ 1 = exp ( S I exp 0J)) Z 0 δj (x) 2 J=0 (J, G. Graphical interpretation: No external points. Vacuum graphs. These terms cancel out in the graphical representation of Green s functions. Cancel out only vacuum graphs, but not all internal loops (divergences)
38 This divergences imply that the functional Z [J] is not well-defined. Regularization allows us to evaluate this functional integral, separating the divergent part. Renormalization absorbs the divergences by a redefinition of the parameter of the theory. The renormalized values of the parameters are considered as the physical ones. Renormalizable theory: at every order in perturbation theory, only a finite number of parameters needs to be renormalized.
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