Download Euclidean path integral formalism: from quantum

Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Euclidean path integral formalism: from quantum
mechanics to quantum field theory
Enea Di Dio
Dr. Philippe de Forcrand
Tutor: Dr. Marco Panero
ETH Zürich
30th March, 2009
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Introduction
Path integral formalisms in quantum mechanics
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Euclidean rotation
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Path integral formalism in quantum field theory
Bosonic field theory
Interacting field
Connection with perturbative expansion
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Introduction
Path integral formalisms in quantum mechanics
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Euclidean rotation
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Path integral formalism in quantum field theory
Bosonic field theory
Interacting field
Connection with perturbative expansion
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Classical mechanics
◮
Lagrangian mechanics. Principle of least action:
δS[L] = 0.
◮
Hamiltonian mechanics. Hamilton’s canonical equations
q̇ =
∂H(q, p)
,
∂p
ṗ = −
Enea Di Dio
∂H(q, p)
.
∂q
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Canonical quantization
◮
Classical variables, Poisson brackets:
{qi , pj } = δij
◮
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.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Introduction
Path integral formalisms in quantum mechanics
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Euclidean rotation
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Path integral formalism in quantum field theory
Bosonic field theory
Interacting field
Connection with perturbative expansion
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Real time
◮
Pictures:
◮
◮
◮
◮
Schrödinger
Heisenberg
Interaction
Time evolution, Schrödinger equation
i
∂ψ(x, t)
= Hψ(x, t)
∂t
|ψ(t)i = e −iH(t−t0 ) |ψ(t0 )i
◮
Probability amplitude for a particle to move from y to x
within time interval t:
hx| e −iHt |y i
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
Free particle, H ≡ H0 =
hx| e −iH0 t |y i =
◮
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
~p 2
2m
m
m 1
2
exp i (x − y )2
2πit
2t
Particle in a potential, H = H0 + V (x), define:
Uǫ ≡ exp (−iHǫ) ∼
= Wǫ ,
m 1
m
ǫ
2
exp i (x − y )2 − i (V (x) + V (y ))
2πiǫ
2ǫ
2
Wǫ as the approximated time evolution operator if ǫ = Nt is
small, because
hx| Wǫ |y i =
◮
exp (−i (H0 + V ) t) = lim Wǫ N .
N→∞
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Inserting N − 1 complete sets of position eigenstates:
Z
E
D x e −iHt y = lim
dx1 · · · dxN−1 hx| Wǫ |x1 i · · · hxN−1 | Wǫ |y i.
N→∞
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
◮
Rewrite as
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
E Z
D x e −iHt y = Dxe iSǫ ,
where
m N
2
dx1 · · · dxN−1
N→∞ 2πiǫ
Dx = lim
◮
and
m
(x − x1 )2 + . . . + (xN−1 − y )2
2ǫ
1
1
V (x) + V (x1 ) + . . . + V (xN−1 ) + V (y )
−ǫ
2
2
Sǫ =
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Sǫ → S, the action:
S=
Z
t
dt ′
0
m
2
ẋ 2 − 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) =
Z
dq ′′ G (q ′ , t ′ ; q ′′ , t ′′ )G (q ′′ , t ′′ ; q, t)
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Euclidean 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:
hx| e
◮
where
SE =
−Hτ
Z
|y i =
t
dτ ′
0
is the Euclidean action.
Enea Di Dio
Z
m
2
Dxe −SE ,
ẋ 2 + V (x)
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
◮
Every path is weighted by e −SE .
◮
Action
S=
Z
t
Z
t
dt ′
0
and Euclidean action
SE =
m
dτ ′
0
2
ẋ 2 − V (x)
m
2
ẋ 2 + V (x)
are related by
S|t=−iτ = iSE ,
substitution
d
dt
d
= i dτ
and dt ′ = −idτ ′ .
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
◮
◮
◮
◮
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Classical mechanics: The path is given by δSE = 0
Quantum mechanics: All paths are possible.
The paths which minimize SE , this means near δSE = 0, are
the most likely.
Physical units: exp (−SE /~)
Classical limit: ~ → 0 ⇒ least action.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Vacuum’s expectation values
◮
◮
Goal: Elegant expression for h0| A |0i.
Let |ni be the energy eigenstates, then
◮
∞
X
e −En τ hn| A |ni
Tr e −Hτ A =
n=0
◮
and
∞
X
e −En τ
Z (τ ) = Tr e −Hτ =
n=0
◮
For τ → ∞ the term E0 dominates the sums, this means
Tr e −Hτ A
h0| A |0i = lim
τ →∞
Z (τ )
◮
Mean in a canonical statistical ensemble.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Example: Correlation functions
◮
hx(t1 ) · · · x(tn )i ≡ h0| x(t1 ) · · · x(tn ) |0i
◮
Continued analytically to Euclidean times τk = itk
◮
1
hx(t1 ) · · · x(tn )i = lim
τ →∞ Z (τ )
where
Z (τ ) =
Z
Z
Dx x (τ1 ) · · · x (τn ) exp (−SE [x (τ )])
Dx exp (−SE [x (τ )]).
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
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.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Introduction
Path integral formalisms in quantum mechanics
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Euclidean rotation
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Path integral formalism in quantum field theory
Bosonic field theory
Interacting field
Connection with perturbative expansion
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Why quantum field theory?
◮
Quantum mechanics and special relativity ⇒ problems
◮
◮
◮
Negative energy states (E 2 = p 2 + m2 )
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.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
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 (x1 , . . . , xn ) = h0| φ (x1 ) · · · φ (xn ) |0i
◮
Wightman functions can be continued analytically into a
region of the complex plane
◮
Define the Schwinger functions as
S . . . ; ~xk , xk4 ; . . . ≡ W . . . ; −ixk4 , ~xk ; . . . ,
for x14 > x24 > . . . > xn4 .
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
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 (x1 , . . . , xn ) =
lim
ǫk →0,ǫk −ǫk+1 >0
Enea Di Dio
S . . . ; ~xk , ixk0 + ǫk ; . . .
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
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.
(Path integral formalism in quantum field theory)
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Time-ordered Green functions
◮
We define the Time-ordered Green functions as
τ (x1 , . . . , xn ) = h0| T φ (x1 ) · · · φ (xn ) |0i ,
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.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Wick rotation
◮
Counter-clockwise rotation
◮
Generalization:
τ (x1 , . . . , xn ) = limπ S . . . ; ~xk , e iφ xk0 ; . . .
φ→ 2
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Bosonic field theory
Interacting field
Introduction
Path integral formalisms in quantum mechanics
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Euclidean rotation
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Path integral formalism in quantum field theory
Bosonic field theory
Interacting field
Connection with perturbative expansion
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Bosonic field theory
Interacting field
Path integral formalism in quantum field theory
◮
Symmetry of Euclidean correlation functions
⇒ Euclidean fields commute
◮
◮
◮
Like classical fields.
Random variables, not operators
Expectation values:
hF [φ]i =
◮
dµF [φ]
For Euclidean functional integral:
dµ =
◮
Z
1 −S[φ] Y
e
dφ (x)
Z
x
Combining, like the Euclidean path integral formalism.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Bosonic field theory
Interacting field
Bosonic field theory
◮
Goal: Find explicitly the expression for the Euclidean
correlation hφ (x1 ) · · · φ (xn )i in Euclidean functional integral
formalism.
◮
From the Gaussian integrals we find
Z
1
1
1
k
−1
d φ exp − (φ, Aφ) + (J, φ) = exp
J, A J
Z0 (J) ≡
Z0
2
2
where J is an arbitrary vector and
Z
1
k
Z0 ≡ d φ exp − (φ, Aφ) .
2
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
Bosonic field theory
Interacting field
For a free field we have
Z
1
4
4
Z [J] ≡ Z0 [J] = exp
d xd yJ (x) G (x, y ) J (y ) ,
2
◮
where G (x, y ) is the propagator that satisfies
+ m2 G (x, y ) = δ (x − y ) .
The correlation functions are given by
δ n Z [J]
.
hφ (x1 ) · · · φ (x2n )i =
δJ (x1 ) · · · δJ (xn ) J=0
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
We move from a discrete to a continuous representation, this
means
◮
◮
◮
Bosonic field theory
Interacting field
φi → φ (x) ,
δ
∂
∂Ji → δJ(x) .
Comparing
1
Z0 [J] = exp
(J, GJ) ,
2
1
J, A−1 J ,
Z0 [J] = exp
2
we find A → G −1 = + m2 .
◮
1
1
1
φ, G −1 φ =
φ, + m2 φ = S0 [φ] .
(φ, Aφ) →
2
2
2
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
Bosonic field theory
Interacting field
Inserting in the Gaussian integrals we find
Z Y
1
Z0 [J] =
dφ (x) exp (−S0 + (J, φ)),
Z0
x
where
Z0 =
Z Y
x
◮
◮
1
2
dφ (x) exp − φ, + m φ .
2
Infinite-dimensional integral ⇒ not well defined.
Define the measure
1
D [φ] e −S0 (φ) ,
dµ0 (φ) =
Z0
where
D [φ] ≡
Y
dφ (x).
x
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
Bosonic field theory
Interacting field
Using this functional approach we find the correlation
functions
Z
hφ (x1 ) · · · φ (xn )i = dµ0 (φ) φ (x1 ) · · · φ (xn ) =
Z Y
1
dφ (x) e −S0 [φ] φ (x1 ) · · · φ (xn ).
Z0
x
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Bosonic field theory
Interacting field
Interacting field
◮
◮
We neglect the problems associated with divergences and
renormalization.
Euclidean action
S [φ] = S0 [φ] + SI [φ] ,
◮
where SI describes the interaction part.
From Gaussian functional integral we find
Z Y
1
Z [J] =
dφin (x) e −S[φ]+(J,φ)
Z
x
where
Z=
Z Y
dφin (x) e −S[φ] ,
x
and φin is the free field.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Introduction
Path integral formalisms in quantum mechanics
Real time
Euclidean time
Vacuum’s expectation values
Euclidean space-time
Euclidean rotation
Why quantum field theory?
Wightman and Schwinger functions
Time-ordered Green functions
Wick rotation
Path integral formalism in quantum field theory
Bosonic field theory
Interacting field
Connection with perturbative expansion
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Connection with perturbative expansion
◮
Goal: Deduce the rules of perturbation theory from functional
integral.
(Feynman rules)
◮
Consider a scalar field with a quartic self-interaction given by
Z
g
SI [φ] =
d 4 x φ (x)4 .
4!
◮
We start to expand e −SI in the functional expression for an
interacting field.
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
We find the functional integral
Z0
1
δ
Z [J] =
exp SI
(J, G0 J) .
exp
Z
δJ (x)
2
◮
For the associated correlations functions we find
Z0 δ
δ
δ
G (x1 , . . . , xn ) =
...
exp SI
Z δJ (x1 )
δJ (xn )
δJ (x)
1
.
· exp
(J, G0 J) 2
J=0
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
Graphical interpretation of the last equation:
h
i
δ
◮ Expand the exponential of SI
δJ(x)
◮
◮
◮
δ
Each derivative δJ(x
is indicated by an external point from
i)
which emerges a line
4
R
δ
Each factor −g d 4 x δJ(x
is indicated by an internal
i)
vertex, from which emerges four lines.
Expand exp
◮
1
2
(J, G0 J)
Internal lines
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
◮
We can normalize the generating functional to Z (0) = 1,
using
Z
1
δ
.
= exp −SI
(J, G0 J) exp
Z0
δJ (x)
2
J=0
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)
Enea Di Dio
Euclidean path integral formalism
Introduction
Path integral formalisms in quantum mechanics
Euclidean rotation
Path integral formalism in quantum field theory
Connection with perturbative expansion
◮
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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Euclidean path integral formalism