notes on quantum field theorypiccinin/qft/notes.pdf1 introduction this lecture notes are for the...
TRANSCRIPT
-
Notes on Quantum Field Theory
Draft of March 20, 2020
Lectures
Fulvio Piccinini
-
Disclaimer: the material contained in these notes (still work in progress) is taken from the
textbooks and lectures notes quoted in the bibliography. It has been written and updated during
the lectures held in previous academic years, starting from 2012-2013. It is used as supporting
material for the lectures on Quantum Field Theory, held at the University of Pavia, a.a. 2019-
2020.
-
Contents
1 Introduction 3
1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Natural units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Relativistic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Brief summary: single particle wave equations . . . . . . . . . . . . . . . . . 6
1.2.1 The Klein Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Canonical field quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 The real Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 The complex Klein Gordon field . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 The Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Propagators, Green functions and causality in quantum field theory . . . . . 16
2 Feynman path-integral quantization in quantum mechanics 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 The propagator as the Green function of the Schrödinger equation . . . . . . 22
2.3 Temporal evolution in position representation . . . . . . . . . . . . . . . . . 24
2.3.1 Infinitesimal time evolution . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Path-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Mathematical difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Euclidean Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 The classical limit and the semiclassical approximation . . . . . . . . 32
2.5.2 The free particle propagator . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.3 Particle in one dimension with generic potential . . . . . . . . . . . . 35
2.5.4 Periodical paths in Euclidean Time . . . . . . . . . . . . . . . . . . . . 38
2.6 Feynman path integral for Euclidean Green Fuctions . . . . . . . . . . . . . 39
2.7 Inverse Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.8 Green functions for the forced harmonic oscillator . . . . . . . . . . . . . . . 47
2.8.1 Meaning of the function D(t) . . . . . . . . . . . . . . . . . . . . . . . 49
-
4 Contents
2.8.2 Functional derivatives of the Groundstate-to-Groundstate transi-
tion amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.8.3 Appendix A: Gaussian Integrals for ordinary functions . . . . . . . . 53
2.8.4 Appendix B: the functional derivative . . . . . . . . . . . . . . . . . . 54
3 Functional quantization for free fields 57
3.1 The scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 The generating functional Z[J] . . . . . . . . . . . . . . . . . . . . . . 57
3.1.2 The Generating Functional Z0E for the free scalar field . . . . . . . . 59
3.1.3 The Generating Functional Z0 for the free scalar field . . . . . . . . . 61
3.1.4 Translation invariance and four-momentum conservation . . . . . . 63
3.2 The Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.1 Introduction: the Fermi-Dirac oscillator . . . . . . . . . . . . . . . . . 64
3.2.2 Grassman algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.3 Grassman functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.4 The Generating Functional for the free Dirac field . . . . . . . . . . . 72
3.2.5 Green functions for the Dirac field . . . . . . . . . . . . . . . . . . . . 74
3.3 The Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.1 Propagator and gauge fixing . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.2 The Generating Functional for the free electromagnetic field . . . . . 79
3.3.3 The Faddeev and Popov method . . . . . . . . . . . . . . . . . . . . . 79
3.4 Appendix A: useful integrals with Grassman variables . . . . . . . . . . . . 84
4 Interacting fields 91
4.1 Perturbative evaluation of Green functions . . . . . . . . . . . . . . . . . . . 91
4.1.1 The Normalization of the Z functional (for scalar fields) . . . . . . . 92
4.1.2 The functional W [J] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.1.3 The Effective Action Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 The S matrix and its relation with Green functions . . . . . . . . . . . . . . . 105
4.2.1 “in” and “out” states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 The S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.3 The optical theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.4 The asymptotic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.5 The Källen-Lehmann spectral representation . . . . . . . . . . . . . . 111
4.2.6 The Lehmann-Symanzyk-Zimmermann reduction formulae . . . . . 113
5 Renormalization 121
5.1 The λϕ4 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Ultraviolet divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3 Power counting for the λϕ4 model . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Regularization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5 Dimensional regularization scheme . . . . . . . . . . . . . . . . . . . . . . . . 127
-
Contents 1
5.6 Calculation of divergent Green functions in λϕ4(x) . . . . . . . . . . . . . . 132
5.6.1 Feynman parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.7 Loop expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.8 One loop renormalization of the λϕ4(x) model . . . . . . . . . . . . . . . . . 136
5.8.1 Bare perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.8.2 Renormalized perturbation theory . . . . . . . . . . . . . . . . . . . . 141
6 QED radiative corrections 147
6.1 Power counting in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2 The generating functionals for QED . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.1 Functional form of the Ward-Takahashi Identity . . . . . . . . . . . . 150
6.2.2 Ward-Takahashi Identities . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.3 Renormalization of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3.1 Ward Identities for renormalized Green functions . . . . . . . . . . . 157
6.3.2 On-shell renormalization scheme in QED . . . . . . . . . . . . . . . . 158
6.4 One-loop radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.4.1 Photon vacuum polarization . . . . . . . . . . . . . . . . . . . . . . . 162
6.4.2 Electron self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.4.3 Explicit calculation of the Ward identity at one-loop . . . . . . . . . . 169
6.4.4 QED counterterms in the on-shell renormalization scheme . . . . . . 170
6.5 The anomalous magnetic moment of the electron . . . . . . . . . . . . . . . . 176
6.6 Infrared divergencies in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.6.1 Soft bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.6.2 Kinoshita-Lee-Nauenberg theorem at O(α): example of cancella-tion of infrared divergencies . . . . . . . . . . . . . . . . . . . . . . . 176
7 The renormalization group 177
Bibliography 183
-
1
Introduction
This lecture notes are for the Quantum Field Theory course of the University of Pavia.
It is a one-semester course and it is meant to follow and complete the course on QED.
In that course you have learned how to quantize a field with the canonical formalism:
the classical fields are interpreted as operators and commutation relations are imposed
on the latters: commutation for fields describing bosonic fields and anticommutation for
fields describing fermionic fields. Then you have seen the formalism applied to QED,
i.e. fermionic and vectorial fields coupled by interaction. In presence of interaction only
approximate methods can be adopted and you have developed QED at first order in
perturbation theory, together with the powerful method of Feynman diagrams. In the
end you were able to calculate several scattering processes cross sections in QED at tree
level. Written in a sentence, the final goal of the present course is to get familiar with
next term in the perturbative expansion in QED , i.e. in the up to one-loop approxima-
tion. Before entering the discussion of the loop diagrams, it is worth mentioning that the
canonical method of field quantization becomes unpractical as soon as you move from
QED to more involved field theories, such as non-abelian gauge theories, like QCD, the
theory of strong interactions and the electroweak Standard Model, which are able to de-
scribe high energy data with high precision. These difficulties can be circumvented by
adopting an alternative approach to quantization, developed by R. Feynman, following
an idea of Dirac: the path-integral approach. This will be illustrated at the beginning in
some detail for ordinary quantum mechanics, to move in the following to the bosonic
fields, treated with functional integral methods. The Feynman rules will be rederived
within this new formalism. The interaction will be introduced first for the case of the λφ4
-
4 1. Introduction
model, because the problems connected with the loop diagrams are already present here,
without additional complications. We will see that, as soon as we try to calculate loop
Feynman diagrams, we encounter the problem of ultraviolet divergences: the integrals
are divergent for large momenta. Physical results can only be obtained after adopting a
regularization procedure and carrying on the renormalization program of redefinition of
parameters and fields. We will discuss in detail the dimensional regularization scheme
and apply it to the λφ4 model. Then we will move to QED, where we will see a powerful
and elegant way of quantize the theory in the presence of gauge invariance. Concerning
the fermionic part, we will go back to the beginning and develop the functional integral
formalism through the use of Grassman variables. At this point we are ready to see the
calculation of loop diagrams in QED. The first example will be the calculation of the O(α)contribution to the electron anomalous moment, which is one of the most extraordinary
tests of QED. Then we will see the complete one-loop renormalization program for QED
and the calculation of a scattering process at one-loop order. The final part of the course
will cover the renormalization group and its link to statistical mechanics.
1.1 Conventions
1.1.1 Natural units
In the c.g.s. system, the fundamental quantities are mass (M), length (L) and time (T).
The system of natural units takes as fundamental quantities mass (M), action (A) and
velocity (V), with units of action and velocity h̄ and c, respectively. So in natural units
h̄ = 1
c = 1 .(1.1)
Since
A = ET = FLT = LMT−2LT = MVL , (1.2)
L =A
MV
T =A
MV2,
(1.3)
a quantity that in c.g.s. has dimensions
MpLqTr = Mp−q−rAq+rV−q−2r , (1.4)
in natural units has dimensions
Mp−q−r . (1.5)
Many quantities have the same dimensions with natural units, e.g. energy, mass, momen-
tum have dimension M, because
E2 = m2 + |~p|2 = m2 + |~k|2 . (1.6)
-
1.1 Conventions 5
Another relevant example is the fine structure constant
α =e2
4πh̄c≃ 1
137(c.g.s) (1.7)
in natural units becomes dimensionless
α =e2
4π≃ 1
137(n.u.) . (1.8)
In order to convert the dimension of a quantity in n.u. to c.g.s. it is enough to multiply
by powers of h̄ and c, respecting Eq. (1.4). The relevant numerical conversion factors are
h̄ = 6.58 × 10−22 MeV · sh̄c = 1.973 × 10−11 MeV · cm .
(1.9)
1.1.2 Relativistic notation
A (contravariant) four-vector in Minkowsky space is denoted as
aµ = (a0 , a1, a2 , a3) = (a0 ,~a) (1.10)
The coordinate vector in c.g.s. units would be
xµ = (t/c, ax , ay, az)|c.g.s. (1.11)
but in natural units it is simply
xµ = (t, ax , ay , az). (1.12)
The metric tensor is
gµν =
1 0 0 0
0 −1 0 00 0 −1 00 0 0 −1
. (1.13)
Note that
gµλgλν = δµν . (1.14)
Covariant vectors are
aµ = gµνaν = (a0 ,−a1 ,−a2,−a3) = (a0 ,−~a), (1.15)
Scalar products are
a · b = aµbµ = a0b0 −~a ·~b. (1.16)Lorentz transformations will be denoted by Lµν. The effect of a Lorentz transforma-
tion is to modify the vectors in this way
aµ −→ a′µ = Lµνaν . (1.17)
-
6 1. Introduction
Lorentz transformations leave scalar products unchanged
aµaν = a′µa′µ = L
µνaνLµ
σaσ
⇒ LµνLµσ = δσν⇒ LµνLµσ = δνσ
(1.18)
Important: the four-dimensional gradient operator is defined as
∂µ =∂
∂xµ=
(
∂
∂x0,
∂
∂x1,
∂
∂x2,
∂
∂x3
)
=(
∂t, ~∇)
. (1.19)
Its tranformation property under a Lorentz transformation is
∂
∂xµ→ ∂
∂x′µ=
∂xν
∂x′µ∂
∂xν=(
Λ−1)µ
ν
∂
∂xν. (1.20)
Even if it is a covariant vector, the relation with the three-dimensional gradient operator
is different from that of a normal vector. This guarantees that the differential of a scalar
function φ(x) is a scalar
δφ(x) = ∂µφ(x)δxµ . (1.21)
The operator
∂µ∂µ = ∂2t − ~∇2 (1.22)
1.2 Brief summary: single particle wave equations
Quantum Field Theory stems from the necessity of reconciling Quantum Mechanics with
the Special Relativity Theory. In fact the latter requires that the laws of physics describing
a physical processes are formally the same in every inertial reference frame. In addition,
all inertial reference frames are related by Lorentz transformations. The cornerstone of
Quantum Mechanics, the Schrödinger equation,
ih̄∂
∂tψ(~x, t) = Hψ(~x, t) , (1.23)
is not invariant under Lorentz tranformations. Space and time variables are not treated
on the same ground.
Remember that the Schrödinger equation (e.g. for a free particle) can be obtained by
taking the classical non-relativistic relation between energy and momentum
E =p2
2m(1.24)
and energy and momentum variables with the corresponding quantum operators
E → i ∂∂t
; ~p → −i~∇ . (1.25)
-
1.2 Brief summary: single particle wave equations 7
1.2.1 The Klein Gordon equation
The first attempt is to apply the substitutions of Eq. (1.25) on the relativistic energy-
momentum relation, which is in natural units (h̄ = 1, c = 1):
E2 = |~p|2 + m2 . (1.26)
This gives rise to the Klein Gordon wave equation
(∂µ∂µ + m2)ψ(x) = 0 . (1.27)
This equation, as a single particle wave equation presents some serious problems:
• both positive and negative energies are allowed. This is not acceptable when we in-troduce interactions because the particle, exchanging energy with the environment,
could pass from positive energy states to arbitrary negative energies, emitting an
infinite amount of energy;
• the temporal component of the conserved current
jµ = (ρ,~j) =i
m[ψ∗∂µψ− (∂µψ∗)ψ] (1.28)
is not positive definite, preventing a probabilistic interpretation (unlike ρ = |ψ|2 forthe Schrödinger equation);
• the second order derivative w.r.t. time conflicts with the principle of quantum me-chanics, according to which the wave function contains all the information on the
state of a physical system and therefore should be completely determined by its
value at the initial time.
1.2.2 The Dirac equation
If the wave function at a certain time must contain all the information on the state, the
wave equation should be of the first order w.r.t. time. On the other hand the relativis-
tic framework requires that time and spatial coordinates appear in a simmetric way, so
that only first order spatial derivatives should appear. On the other hand the solution
of the hypothetic equation should be compatible with the Klein Gordon equation, which
satisfies automatically the relativistic energy-momentum equation. To satisfy the above
requirements, Dirac proposed the following wave equation
i∂ψ
∂t=(
−i~α · ~∇+βm)
ψ , (1.29)
where αi (i = 1, 2, 3) and β are hermitean (since the Hamiltonian is hermitean) matrices
N × N to be determined, and the wave function is a column vector with N components.
-
8 1. Introduction
In order to comply with Eq. (1.29), theαi andβmatrices must obey the following relations
{
αi,α j}
= 2δi j
{αi,β} = 0α2i = β
2 = 1 .
(1.30)
This can be seen by requiring
(~α · ~p +βm) (~α · ~p +βm) = E2|~p|2 + m2 , (1.31)
i.e.
αiα jpi p j + m (αiβ+βαi) pi +β2m2
=1
2
({
αi,α j}
+[
αiα j])
pi p j + m {αi,β} pi +β2m2
= pi p jδi j + m2 .
(1.32)
The third relation of Eq. (1.30) implies that the eigenvalues of matrices αi and β are all
= ±1. From the first and third relations, instead, we have
αi = αiα jα j = −α jαiα jβ = βα jα j = −α jβα j ,
(1.33)
where the repeated indices are not summed. Taking the traces:
Tr (αi) = Tr(
αiα jα j)
= Tr(
α jαiα j)
= −Tr(
α jαiα j)
= −Tr (αi)Tr (β) = Tr
(
βα jα j)
= Tr(
α jβα j)
= −Tr (α jβα j)
= −Tr (β)(1.34)
Since the matrices αi and β have to be traceless and have eigenvalues = ±1, the dimen-sion N can be only even. N = 2 is excluded because the standard 2 × 2 Pauli matricesσi
σ1 =
(
0 1
1 0
)
, σ2 =
(
0 −ii 0
)
, σ3 =
(
1 0
0 1
)
, (1.35)
satisfy the same anticommutation of Eq. (1.30), but they are a basis, together with the
Identity matrix, for the 2 × 2 matrices. So that it is impossible to find a fourth indepen-dent matrix which anticommutes with σi. So the minimum dimensions for the αi and β
matrices is N = 4.
An explicit representation is the following one:
αi =
(
0 σiσi 0
)
, β =
(
1 0
0 −1
)
. (1.36)
-
1.2 Brief summary: single particle wave equations 9
Usually the Dirac equation is written in terms of the γ matrices, defined in the follow-
ing way:
γ0 = β , γi = βαi = γ0αi , (1.37)
which satisfy the anticommutation rules
{γµγν} = 2gµν . (1.38)
The γ matrices have the following properties
(
γ0)2
= 1 , (1.39)(
γi)2
= −1 (1.40)
㵆= γ0γµγ0 . (1.41)
Note that the γ matrices are not hermitean.
A possible explicit representation of the γ matrices is the Dirac-Pauli representation (use-
ful for massive particles and for taking the non relativistic limit)
γ0 =
(
1 0
0 −1
)
, γi =
(
0 σ i
−σ i 0
)
. (1.42)
In terms of the γ matrices the Dirac equation assumes the form
(iγµ∂µ − m1)ψ = 0 . (1.43)
What kind of particles are described by the Dirac equation? If we consider the Hamil-
tonian of Eq. (1.29)
H = ~α · ~p +βm , (1.44)
it is easy to check that it does not commute with the angular momentum operator ~L =
~x × ~p. E.g.
[H, L3] = [αi pi, x1 p2 − p2x1] = α1p2 [p1, x1]−α2p1 [p2, x2] = −i (α1p2 −α2p1) . (1.45)
However, let us consider the operator
~Σ =
(
~σ 0
0 ~σ
)
, (1.46)
-
10 1. Introduction
which, heuristically, can be interpreted as the extension to four dimensions of the quan-
tum mechanical spin operator, represented by the Pauli matrices 1. By defining the anti-
symmetric tensor
σµν ≡ i2[γµ ,γν] , (1.47)
its spatial components are
σ i j =i
2
[
γi,γ j]
. (1.48)
Using the Dirac-Pauli representation of the γ matrices, and considering that
(
0 σ i
−σ i 0
)(
0 σ j
−σ j 0
)
=
(−σ iσ j 00 −σ iσ j
)
, (1.49)
we get
σµν =i
2[γµ ,γν]− i
2
([
σ i,σ j]
0
0[
σ i,σ j]
)
= εi jk(
σk 0
0 σk
)
= εi jkΣk . (1.50)
From Eq. (1.50) we can write
σ12 = ε123Σ3
σ21 = ε213Σ3 = −ε123Σ3 .(1.51)
From the above equation and form Eq. (1.48) we can write
σ21 −σ21 = 2Σ3 =i
2
[
γ1,γ2]
− i2
[
γ2,γ1]
= i[
γ1,γ2]
, (1.52)
i.e., using Eq. (1.37)
Σ3 =i
2[γ1,γ2] =
i
2
[
γ0α1,γ0α2
]
= − i2(α1α2 −α2α1) . (1.53)
Remembering Eq. (1.45), let us calculate the following commutator of Σ3 with the Hamil-
tonian
1
2[H,Σ3] = −
i
4[αi pi,α1α2 −α2α1] = −
i
2[αi pi ,α1α2]
= − i2[α1p1,α1α2]−
i
2[α2p2,α1α2]
= − i2
(
α21α2 −α1α2α1)
p1 −i
2
(
α2α1α2 −α1α22)
p2
= −i (α2p1 −α1p2) = − [H, L3] .
(1.54)
1Actually the correct relativistic operator is given by the Pauli-Lubanski operator Wµ = − 12εµνρσ JνρPσ ,where Jµν is the angular momentum operator, generator of rotations, and Pµ is the generator of the spatial
traslations. For a particle of mass m we can perform a Lorentz transformation to go to the rest frame, where
the only contribution is σ = 0
-
1.2 Brief summary: single particle wave equations 11
By defining the total angular momentum as
~J ≡ ~L + 12~Σ , (1.55)
we have
[H, J3] =
[
H, L3 +1
2Σ3
]
= 0 , (1.56)
i.e. 12Σ3 is the third component of the spin of of the particle described by the Dirac equa-
tion. Since the eigenvalues of Σ3 are ±1, the Dirac equation describes particles with spin12 .
Eq. (1.54) shows that, when the momentum ~p is different from zero, the projection of spin on a
generic axis is not a constant of the motion.
An important quantity is the projection of the spin on the momentum direction, known
as the helicity
σp =~Σ · ~p|~p| . (1.57)
which is a constant of the motion, i.e. it commutes with the Hamiltonian:
[
H,σp]
=1
|~p|[
αi pi +βm,Σ j p j]
=1
|~p|[
αi pi,Σ j p j]
=1
|~p|[
αi,Σ j]
pi p j . (1.58)
From Eq. (1.53), by permutation of the indices we have
Σ1 = −i
2(α2α3 −α3α2) = −iα2α3
Σ2 = −i
2(α3α1 −α1α3) = −iα3α1
Σ3 = −i
2(α1α2 −α2α1) = −iα1α2 .
(1.59)
In Eq. (1.58), the three terms of the form [αi,Σi] are 0 because they are of the form −i(αiα jαk −α jαkαi) = 0 by anticommutation. Then we have the following six terms
[α1,Σ2] = −i [α1,α3α1] = +2iα3[α1,Σ3] = −i [α1,α1α2] = −2iα2[α2,Σ1] = −i [α2,α2α3] = −2iα3[α2,Σ3] = −i [α2,α1α2] = +2iα1[α3,Σ1] = −i [α3,α2α3] = +2iα2[α3,Σ2] = −i [α3,α3α1] = −2iα1 .
(1.60)
So the terms[
αi,Σ j]
are antisymmetric w.r.t. exchange of i and j. Since they are multiplied
(and summed over i and j) by the symmetric term pi p j, we can conclude[
H,σp]
= 0 . (1.61)
So in the end, the solutions of the Dirac equation are four-component spinors, describing
states with spin 12 and positive and negative energy.
-
12 1. Introduction
1.3 Canonical field quantization
Another heuristic argument conflicting with the hypothesis of a wave function for a rel-
ativistic particle is the following: if we try to measure the position of an electron with
a microscope, we have to use light with wave length λ (the greater the precision, the
smaller λ). On the other hand the electron receives a random momentum of the order of
the momentum of the photon k = h̄λ
. The indeterminacy on the position and momentum
must satisfy the indetermination principle
∆x ∼ λ = h̄∆p
. (1.62)
When the energy of the photon is greater than the threshold 2me we can have the creation
of a pair e+e−, so that the concept of position of the electron becomes ambiguous sincewe have two electrons.
In essence, our theoretical formulation should be able to describe processes with variable
number of particles.
The only way out is to abandon the idea of single particle wave equation and consider
that the equation (Klein-Gordon or Dirac) describes a (classical) field. In this way we are
going to pass from a system with a discrete number of degrees of freedom to a continuum
system, with an infinite number of degrees of freedom. In the field theory approach,
space ~x and time t are parameters and the value of the field φ(~x, t) is the dynamical
variable (the analogous role played by ~x(t) in non-relativistic quantum mechanics).
Actually the fields are distributions and not observables. The latter are bilinear in the
fields.
1.3.1 The real Klein-Gordon field
The classical field theory is defined when we spcify the Lagrangian density L. For thecase of the real scalar field, we have
L = 12
(
∂µφ∂µφ− m2φ2
)
, (1.63)
which gives the Klein-Gordon equation as Euler-Lagrange equation, from the require-
ment of minimum variation of the Action S =∫
d4xL(φ, ∂µφ). Since the Lagrangiandensity is Lorentz-invariant and has no explicit dependence on x, the Noether theorem
guarantees the conservation of four-momentum and angular momentum. The energy-
momentum tensor is
Tµν = ∂µφ∂νφ− gµνL . (1.64)The (classical) energy of the field is given by the space integral of T00:
H =∫
T00 d3x =1
2
∫
[
(∂0φ)2 + ~∇φ · ~∇φ+ m2φ2
]
. (1.65)
-
1.3 Canonical field quantization 13
From Eq. (1.65), we can see that the scalar field, whose equation of motion is the Klein-
Gordon equation, is not affected by the negative energy problem.
Up to now we don’t have yet a connection between the field and particles. This is
achieved through the field quantization procedure, named also second quantization:
the field is regarded as an Hermitean linear operator (in the Heisenberg representation),
which satisfies canonical equal time commutation relations, analogously to the commu-
tation relations of ordinary quantum mechanics ([xi, p j] = iδi j).
It is convenient to use the following Fourier representation
φ(x) = φ+(x) +φ−(x) = ∑~k
1√2V2ωk
[
a(k)e−ikx + a†(k)e+ikx]
, (1.66)
withω2k = |~k|2 + m2, as imposed by the equation of motion.Observations:
• φ+ contains the positive frequencies whileφ− contains the negative frequencies;
• a(k) and a†(k) are operators which inherit the commutation rules from the field andits canonically conjugate momentum.
The canonically conjugate momentum is given by
π(x) =∂L
∂ (∂0φ). (1.67)
The equal time commutation relations are
[
φ(~x, t), π(~x′ , t)]
= iδ(3)(~x −~x′)[
φ(~x, t),φ(~x′ , t)]
=[
π(~x, t), π(~x′ , t)]
= 0 .(1.68)
Eq. (1.68) implies the following commutation relations for the ladder operators a and a†
[
a(k), a†(k′)]
= δ~k~k′[
a(k), a(k′)]
=[
a†(k), a†(k′)]
= 0 .(1.69)
The above commutation relations are exactly the ones of the harmonic oscillator. The
scalar quantum field is equivalent to a collection of harmonic oscillators, one for each
mode~k. We can define the operator
N(~k) = a†(~k)a(~k) . (1.70)
-
14 1. Introduction
with the properties
[
N(~k), a†(~k)]
= a†(~k)[
N(~k), a(~k)]
= −a(~k) .(1.71)
We can take the eigenstates of the operator N,∣
∣n(~k)〉
and verify that if∣
∣n(~k)〉
has eigen-
value n(~k), the states a†(~k)(~k) and a(~k)(~k) have eigenvalues n(~k)+ 1 and n(~k)− 1, respec-tively.
In terms of ladder operators, the Hamiltonian of the field of Eq. (1.65) becomes
H =1
2 ∑~k
ωk
(
a(~k)a†(~k) + a†(~k)a(~k))
= ∑~k
ωk
(
N(~k) +1
2
)
, (1.72)
and the momentum of the field
Pi =∫
d3x T0i =∫
d3x ∂0φ∂iφ =
1
2 ∑~k
ki(
a(~k)a†(~k) + a†(~k)a(~k))
= ∑~k
ki(
N(~k) +1
2
)
.
(1.73)
Eqs. (1.72,1.73) suggest that the operator N(~k) is the number operator of particles with
momentum ~k and energy ω~k, provided the eigenvalues n(~k) never become negative.
This can be shown by looking at the norm of the generic state a(~k)∣
∣n(~k)〉
:
[
a(~k)∣
∣n(~k)〉
]† [a(~k)
∣
∣n(~k)〉
]
=〈
n(~k)∣
∣a†(~k)a(~k)∣
∣n(~k)〉
= n(~k)〈n(~k)∣
∣n(~k)〉
> 0 .(1.74)
The Hilbert space on which the field operator act is given by the tensorial product of the
states of the different oscillators. To this space belong:
• the vacuum state∣
∣0〉
, where all oscillators are in the ground state; the vacuum state
is determined by the condition of being annihilated by the destruction operator
a(~k)∣
∣0〉
= 0
• states with different excitation numbers of the various oscillators, obtained by actingon the vacuum with the creation operators a†
∣
∣n1, n2, . . .〉
=1√
n1!n2!, . . .
[
a†(~k1)]n1[
a†(~k2)]n2
. . .∣
∣0〉
. (1.75)
Since there is no limit to the occupation numbers, the real scalar field describes a system
of bosonic particles. This is a direct consequence of the canonical commutation rela-
tions.
-
1.3 Canonical field quantization 15
The step from the classical observables (e.g. energy, momentum) to the quantum ver-
sions has an intrinsic ambiguity, since we transform products of commuting variables
into products of non-commuting operators. The consequence is for instance that we have
an infinite energy, given by the sum of all ground state energies of the oscillators. Since
we are free to fix the zero energy level, we can remove this infinity by adopting the Nor-
mal ordering prescription: in each normal ordered product of operators the absorption
operators always stand to the right of creation operators. Example (remember thatφ+(x)
contains the absorption operator):
N [φ(x)φ(y)] =(
φ+(x) +φ−(x)) (
φ+(y) +φ−(y))
= φ+(x)φ+(y) +φ−(y)φ+(x)
= +φ−(x)φ+(y) +φ−(x)φ−(y)
(1.76)
1.3.2 The complex Klein Gordon field
The classical field theory of a complex scalar field is equivalent to the one of two real
scalar fieldsφ1 andφ2. In fact we may consider
φ =(φ1 + iφ2)√
2
φ∗ =(φ1 − iφ2)√
2
(1.77)
Treating φ and φ∗ as two independent fields, the Lagrangian density (which is real, togive a real Action) is
L = (∂µφ) (∂µφ∗)− m2φ∗φ . (1.78)In terms of the real fieldsφ1 andφ2, the Lagrangian density would be
L =(
∂µ~φ)
·(
∂µ~φ)
− m2~φ · ~φ , (1.79)
where ~φ is a two dimensional vector with componentsφ1(x) andφ2(x).
The corresponding Euler-Lagrange equations are two independent Klein-Gordon equa-
tions for φ and φ∗. The treatment is similar to the case of the real scalar field, with twodegrees of freedom instead of one. However, the Lagrangian density of Eq. (1.78) shows
an additional “internal” simmetry w.r.t. to Eq. (1.63) of the real Klein-Gordon field: the
Lagrangian density is invariant under the transformation
φ(x) → e−iαφ(x)φ∗(x) → eiαφ∗(x) ,
(1.80)
where α is a “global” phase, (global because it does not depend on x). The transforma-
tion ∈ group U(1). Adopting the degrees of freedom φ1 and φ2 the same tranformation
-
16 1. Introduction
would be a rotation in two dimensions (∈ group SO(2)) by an angle α, which mixes thecomponentsφ1 andφ2.
Through the Noether theorem, this symmetry has an associated conserved current
(∂µ Jµ = 0):
Jµ =∂L
∂ (∂µφ)δφ+
∂L∂ (∂µφ∗)
δφ∗ = i (φ∗∂µφ−φ∂µφ∗) . (1.81)
The corresponding conserved quantity is the space integral of the temporal component
J0
Q =∫
J0 d3x = i∫
(
φ∗∂φ
∂t−φ∂φ
∗
∂t
)
. (1.82)
At this point Q is a purely classical charge. The connection with the electromagnetic
charge can be seen in the following way: if we perform a global transformation we are
propagating a faster than light signal, which is not compatible with special relativity.
Instead we should admit the possibility of changing the phase locally. In this way the
Lagrangian density of Eq. (1.78) is not invariant, but for the time being let us stay with
the global tranformation.
Observation: Charged scalar fields can only be described by complex scalar fields. For a
real field the conserved current does not exist, Jµ = 0.
The quantization proceeds in the same way as for the real scalar field, but the quantum
field is not Hermitean anymore:
φ(x) = φ+(x) +φ−(x) = ∑~k
1√2V2ωk
[
a(k)e−ikx + b†(k)e+ikx]
φ†(x) = φ†+(x) +φ†−(x) = ∑~k
1√2V2ωk
[
b(k)e−ikx + a†(k)e+ikx]
,
(1.83)
session to be continued
1.3.3 The Dirac field
session to be written
1.4 Propagators, Green functions and causality in quantum
field theory
In the canonical formalism, the propagator is defined as the vacuum expectation value
of the time-ordered product of two field operators. For instance, the propagator for the
scalar field is
i∆(x, y) =〈
0∣
∣T(
ϕ(x)ϕ†(y))
∣
∣0〉
, (1.84)
-
1.4 Propagators, Green functions and causality in quantum field theory 17
i.e.
i∆(x, y) =〈
0∣
∣ϕ(x)ϕ†(y)∣
∣0〉
for x0 > y0 ,
i∆(x, y) =〈
0∣
∣ϕ†(y)ϕ(x)∣
∣0〉
for y0 > x0 .
It gives the quantum amplitude for the creation of a quantum by a source localized at
y and its absorption in x, if x0 > y0. Instead, if x0 < y0, it describes the creation of
an antiparticle in x and its absorption in y. A general property of the propagator in a
translationally invariant quantum field theory is its dependence on the difference on co-
ordinates. In fact, if we insert in Eq. (1.84) the identity given by the product U†U, where Uis the translation operation of −y and take into account that the vacuum is translationallyinvariant, we get
i∆(x, y) =〈
0∣
∣U†UT(
ϕ(x)U†Uϕ†(y))
U†U∣
∣0〉
=〈
0∣
∣T(
ϕ(x − y)ϕ†(0))
∣
∣0〉
= i∆(x − y, 0) ≡ i∆(x − y) . (1.85)
Introducing the positive and negative frequency parts of the fieldϕ(x)(+) 2 given by 3
ϕ(+)(x) =∫
d3k
(2π)32ωkake
−i(k·x) , (1.86)
ϕ(−)(x) =∫
d3k
(2π)32ωkb†ke
+i(k·x) , (1.87)
we have
i∆(x) =〈
0∣
∣
[
ϕ(+)(x)(
ϕ†)(−)
(0)
]
∣
∣0〉
for x0 > 0 ,
i∆(x) = −〈0∣
∣
[
ϕ(−)(x)(
ϕ†)(+)
(0)
]
∣
∣0〉
for x0 < 0 .
It is useful to introduce the functions ∆(+)(x) and ∆(−)(x), defined by means of the aboveequations:
i∆(+)(x) =〈
0∣
∣
[
ϕ(+)(x),(
ϕ†)(−)
(0)
]
∣
∣0〉
, (1.88)
i∆(−)(x) = −〈
0∣
∣
[
ϕ(−)(x),(
ϕ†)(+)
(0)
]
∣
∣0〉
. (1.89)
2The positive frequency part is the one with temporal evolution e−iωt (and is multiplied by the annihi-lation operator), while the negative frequency part has the temporal evolution e+iωt (and is multiplied by
the creation operator).3Notice that we use here the normalization of continuum.
-
18 1. Introduction
By using Eqs. (1.86) and (1.87), we can verify that ∆(+) and ∆(−) are solution of the Klein-Gordon equation of Eq. (1.27). In particular, using the Cauchy theorem, we have:
i∆(+)(x) =1
(2π)3
∫
d3k
2ωe−i(k·x) , (1.90)
i∆(−)(x) =−1
(2π)3
∫
d3k
2ωe+i(k·x) . (1.91)
At this point it is useful to make a digression on Green functions. We can consider the
equation of motion of the Klein-Gordon field in the presence of a given external source
(−∂µ∂µ − m2)ϕ(x) = J(x) . (1.92)
The Klein-Gordon equation of Eq. (1.27) is the omogeneus equation associated to Eq. (1.92).
The solutions of Eq. (1.92) can be obtained through the Green function of the problem,
which is the solution of the same equation, with the external source given by a Dirac
delta localized at the origin of space-time:
(−∂µ∂µ − m2)G(x) = δ(x) . (1.93)
Knowing G(x), the solution of Eq. (1.92) is given by
ϕ(x) =∫
dy G(x − y)J(y) . (1.94)
The number of Green functions associated with Eq. (1.92) is infinite. A particular Green
function can be singled out fixing the boundary conditions. Eq. (1.93) can be solved
through Fourier transforms:
f̃ (k) =∫
d4x f (x)ei(k·x)
f (x) =1
(2π)4
∫
d4k f̃ (k)e−i(k·x) .
Eqs. (1.93) and (1.94) become
G̃(k) =1
k2 − m2 , (1.95)ϕ̃(k) = G̃(k) · J̃(k) . (1.96)
By means of Eq. (1.96) we can write a formal particular solution of Eq. (1.92):
ϕ(x) =∫
d4k e−i(k·x)1
k2 − m2 J̃(k) . (1.97)
The integration of Eq. (1.97) presents singularities along the real axis in the points cor-
responding to the propagation of free waves: k0 = ±ω. Every particular solution can
-
1.4 Propagators, Green functions and causality in quantum field theory 19
be found by giving a particular path in the complex plane (Re(k0), Im(k0)) to define the
integral. The Feynman prescription is the one for which i∆ = +i∆(+)(x) when t > 0 and
i∆ = −i∆(−)(x) when t < 0. This prescription can be obtained by shifting the singularityat k0 = ω by a negative infinitesimal imaginary part and the one at k0 = −ω by a posi-tive infinitesimal imaginary part. For x0 > 0 we can choose a closed path enclosing the
singularity at k0 = ω (we close the path in the lower semiplane), finding
i∆F(x) = i∆(+)(x) . (1.98)
For x0 < 0 we can choose a closed path enclosing the singularity at k0 = −ω (we closethe path in the upper semiplane), finding
i∆F(x) = −i∆(−)(x) . (1.99)
The general form of the Feynman propagator is
i∆F(x) = ϑ(x0)i∆(+)(x)− ϑ(−x0)i∆(−)(x) . (1.100)
In summary, the Feynman propagator is given by
i∆F(x) =〈
0∣
∣T(
ϕ(x)ϕ†(0))
∣
∣0〉
=∫
d4k
(2π)4i
k2 − m2 + iε e−i(k·x) . (1.101)
The Feynman prescription amounts to give an infinitesimal negative imaginary part to
the mass: m2 → m2 − iε. This prescription fixes the way of handling the singularities onthe real axis imposing that positive frequencies are propagated forward in time and neg-
ative frequencies are propagated backward in time. In fact, for positive time, the path has
to be closed in the lower semiplane and the solution singles out the pole at k0 = +ω; for
negative time, the path has to be closed in the upper semiplane and the solution singles
out the pole at k0 = −ω. This complies with the Feynman-Stueckelberg interpretation:a particle with positive energy is propagated forward in time and a quantum with neg-
ative energy backward in time, i.e. an antiparticle with positive energy forward in time.
We will see that the same prescription will ensure the convergence of the path-integral
formulation of quantum field theory.
-
2
Feynman path-integralquantization in quantum
mechanics
2.1 Introduction
During the QED course you have seen how to quantize field theories with the canoni-
cal quantization approach, with particular reference to QED. Since in this formalism the
manifest Lorentz invariance is broken, it becomes very difficult to apply to more complex
theories, like gauge theories. For this reason an alternative method has been developed
by Feynman, based on an idea of Dirac: the path integral approach. The main idea of the
method is based on the superposition principle, which is at the roots of quantum mechanics. For
this reason we will firstly illustrate the path integral approach for quantum mechanics,
with particular reference to the harmonic oscillator. The importance of this system for
quantum field theory is due to the fact that a bosonic field is dynamically equivalent to a
set of an infinite number of harmonic oscillators (normal modes). We remember that the
classical Hamiltonian of the free oscillator is
Hcl =p2
2m+
1
2kq2 , (2.1)
-
22 2. Feynman path-integral quantization in quantum mechanics
which, upon quantization, becomes the operator
H = − h̄2
2m
d2
dq2+
1
2mω20q
2 . (2.2)
For simplicity of notation, we will subtract to Eq. (2.2) the zero point energy 12 h̄ω0 and
use natural units and m = 1. Hence we have
H = −12
d2
dq2+
1
2ω20q
2 − 12ω0 . (2.3)
The eigenfunctions of H are expressed in terms of the Hérmite polinomials as follows
〈q|n〉 = Φn(q) =ω140π
− 14 2−n2 (n!)−
12 exp
(
−12ω0q
2
)
Hn
(
ω120 q
)
(2.4)
and the Hérmite polinomial of order n is given by
Hn(z) = (−)n exp(
z2)
(
d
dz
)n
exp(
−z2)
. (2.5)
The eigenvalues of H areEn = nω0 (2.6)
2.2 The propagator as the Green function of the Schrödinger
equation
Before entering the details of the Feynman quantization of the harmonic oscillator, we
show that the knowledge of the solution of the Schrödinger equation for a generic non-
relativistic system, the wave function ψ (~q, t), is equivalent to the knowledge of the
Green function of the Schrödinger equation with a particular initial condition. Limit-
ing ourselves to a one-dimensional system, for instance a particle in a one-dimensional
potential, the Schrödinger equation reads
(
ih̄∂
∂t− H
)
ψ(q, t) = 0 , (2.7)
where H = − h̄22m ∂2
∂q2+ V(q). Next we consider the function K(q′ , t′; q, t), also called the
propagator, defined as the solution of the equation
(
ih̄∂
∂t′− Hq′
)
K(q′ , t′; q, t) = ih̄δ(q′ − q)δ(t′ − t) . (2.8)
with initial condition
K(q′ , t; q, t) = δ(q′ − q) . (2.9)
-
2.2 The propagator as the Green function of the Schrödinger equation 23
By means of the Green function K(q′ , t′; q, t), we can write the wave function, solution ofthe Schrödinger equation of Eq. (2.7), as
ψ(q′ , t′) = ϑ(t′ − t)∫
dq K(q′ , t′; q, t)ψ(q, t) . (2.10)
Eq. (2.10) expresses the Huygen’s principle. We can verify that the function of Eq. (2.10)
is solution of Eq. (2.7) by substitution:(
ih̄∂
∂t− H
)
ψ(q, t) = ϑ(t − ti)(
ih̄∂
∂t− H
)
∫
dqi K(q, t; qi , ti)ψ(qi , ti)
= ϑ(t − ti)∫
dqi
(
ih̄∂
∂t− Hq
)
K(q, t; qi , ti)ψ(qi , ti)
= ϑ(t − ti)∫
dqiih̄δ(q − qi)δ(t − ti)ψ(qi , ti)= ih̄δ(t − ti)ψ(q, ti)ϑ(t − ti) = 0 for every t > ti . (2.11)
Thus the ψ defined by Eq. (2.10) is a solution of the Schrödinger equation for all times
t > ti. The restriction on the times preserves causality.
From the second line of Eq. (2.11) we can see that for t′ < t must obey the followingequation:
(
ih̄∂
∂t′− Hq′
)
K(q′ , t′; q, t) = 0 for every t , (2.12)
with the initial condition of Eq. (2.9).
We can find an explicit form for the propagator by means of the solutions of the station-
ary Schrödinger equation ϕn(q) and the corresponding eigenvalues En. Since the ϕn(q)
form a complete system, K(q′ , t′; q, t) can be expanded in this basis:
K(q′ , t′; q, t) = ϑ(t′ − t)∑n
anϕn(q′)e−
ih̄ Ent
′, (2.13)
where we have directly introduced the constraints on forward times. The expansion co-
efficients depend in general on q and t: an = an(q, t). Because of the initial condition of
Eq. (2.9), we have
K(q′ , t; q, t) = δ(q′ − q) = ∑n
an(q, t)ϕn(q′)e−
ih̄ Ent . (2.14)
Since δ(q′ − q) is time-independent, the same is true also for the r.h.s. of Eq. (2.14). Thisimplies that
an(q, t) = an(q)e+ ih̄ Ent , (2.15)
i.e.
δ(q′ − q) = ∑n
an(q)ϕn(q′) , (2.16)
which is fulfilled by
an(q) = ϕ∗n(q) . (2.17)
-
24 2. Feynman path-integral quantization in quantum mechanics
Putting everything together,
K(q′ , t′; q, t) = ϑ(t′ − t)∑n
ϕ∗n(q)ϕn(q′)e−
ih̄ En(t
′−t) , (2.18)
Observing thatϕn(q) =〈
q∣
∣n〉, we can write
K(q′ , t′; q, t) = ϑ(t′ − t)∑n
ϕ∗n(q)ϕn(q′)e−
ih̄ En(t
′−t)
= ϑ(t′ − t)∑n
〈
n∣
∣q〉e− ih̄ En(t′−t)〈
q′∣
∣n〉
= ϑ(t′ − t)∑n
〈
n∣
∣e+ih̄ Ht∣
∣q〉〈
q′∣
∣e−ih̄ Ht
′∣∣n〉
= ϑ(t′ − t)〈
q′∣
∣e−ih̄ H(t
′−t)∣∣q〉
≡ ϑ(t′ − t)〈
q′∣
∣U(t′ , t)∣
∣q〉
. (2.19)
Thus the propagator is the time devolpment operator for t′ > t in position representation.
2.3 Temporal evolution in position representation
Since the Feynman quantization is based on the study of the temporal evolution of the
field in the position representation, we consider now the temporal evolution of the har-
monic oscillator. Given at t = 0 the system in the position eigenstate∣
∣q〉
, let’s evolve the
system for a time t and consider the probability amplitude of finding the system at time t
in the position eigenstate∣
∣q′〉
:
〈
q′∣
∣ exp (−itH)∣
∣q〉
=∞
∑n,m=0
〈q′|m〉〈
m∣
∣ exp (−itH)∣
∣n〉
〈n|q〉
=∞
∑n=0
exp (−inω0t)〈q′|n〉〈n|q〉
=∞
∑n=0
exp (−inω0t)Φn(q′)Φ∗n(q) ,
(2.20)
which is equivalent to Eq. (2.18). In order to calculate Eq. (2.20) for an arbitrary finite
time t, from Eq. (2.4) and Eq.(2.5) we get
Φn(q) = (−)nω140π
− 14 2−n2 (n!)−
12ω
− n20 exp
(
1
2ω0q
2
)(
∂
∂q
)n
exp(
−ω0q2)
. (2.21)
-
2.3 Temporal evolution in position representation 25
Substituting Eq. (2.21) in Eq. (2.20) we get
〈
q′∣
∣ exp (−itH)∣
∣q〉
=(ω0
π
)12
exp
[
1
2ω0
(
q2 + q′2)
]
×∞
∑n=0
1
n!
(
exp (−iω0t)2ω0
∂
∂q′∂
∂q
)n
exp[
−ω0(
q2 + q′2)]
=(ω0
π
)12
exp
[
1
2ω0
(
q2 + q′2)
]
× exp(
exp (−iω0t)2ω0
∂
∂q′∂
∂q
)
exp[
−ω0(
q2 + q′2)]
.
(2.22)
Now it is convenient to express the last gaussian factor in terms of plane waves (eigen-
functions of the position operator) through the following integral representation (which
can be verified by means of Eq. (2.202))
exp(
−ω0q2)
=1
2(πω0)
− 12∫ +∞
−∞dk exp
(
ikq − k2
4ω0
)
(2.23)
By substitution of Eq. (2.23) in Eq. (2.22) we get
〈
q′∣
∣ exp (−itH)∣
∣q〉
=1
4π−
32ω
− 120 exp
[
1
2ω0
(
q2 + q′2)
]
exp
(
exp (−iω0t)2ω0
∂
∂q′∂
∂q
)
∫ +∞
−∞dk′ exp
(
ik′q′ − k′2
4ω0
)
∫ +∞
−∞dk exp
(
ikq − k2
4ω0
)
=1
4π−
32ω
− 120 exp
[
1
2ω0
(
q2 + q′2)
]
∫ +∞
−∞dk∫ +∞
−∞dk′ exp
[
−exp(−iω0t)2ω0
kk′ + ikq + ik′q′ − k2
4ω0− k
′2
4ω0
]
.
(2.24)
The two-dimensional integral of Eq. (2.24) can be solved through Eq. (2.203). In fact the
argument of the exponential can be written in the form
−xT · A · x + bT · x ,
where
~x =
(
k
k′
)
, A =1
4ω0
(
1 γ
γ 1
)
, ~b = i
(
q
q′
)
, γ = exp (−iω0t) . (2.25)
From the above equations we get
det A =1
(4ω0)2(1 − γ2) , A−1 = 1
det A
1
4ω0
(
1 −γ−γ 1
)
. (2.26)
-
26 2. Feynman path-integral quantization in quantum mechanics
The eigenvalues of A are
λ± =1 ± γ4ω0
,
which are positive and allow to apply the n-dimensional gaussian integration formula of
Eq. (2.203). The result of the bidimensional integral in Eq. (2.24) is
∫ +∞
−∞dk∫ +∞
−∞dk′ [. . .] =
4πω0√
1 − γ2exp
[
− ω01 − γ2
(
q2 + q′2 − 2γqq′)
]
, (2.27)
so that Eq. (2.24) becomes
〈
q′∣
∣ exp (−itH)∣
∣q〉
=
[
ω0
π(1 − γ2)
]12
exp
{
− 11 − γ2
[
1
2ω0
(
1 + γ2) (
q2 + q′2)
− 2ω0γqq′]}
.
(2.28)
Observing that
2qq′ =1
2
(
q + q′)2 − 1
2
(
q − q′)2
, (2.29)
we can manipulate the argument of the exponential in Eq. (2.28) in the following way:
− 11 − γ2
[
1
2ω0
(
1 + γ2) (
q2 + q′2)
− 2ω0γqq′]
= − ω01 − γ2
[
1
2
(
1 + γ2) (
q2 + q′2)
− 2γqq′]
= − ω01 − γ2
[
1
2
(
1 + γ2) (
q2 + q′2)
− γ2
(
q + q′)2
+γ
2
(
q − q′)2]
= − ω01 − γ2
{
1
4
(
1 + γ2) [
(
q + q′)2
+(
q − q′)2]
− γ2
(
q + q′)2
+γ
2
(
q − q′)2}
= − ω01 − γ2
{
1
4(1 − γ)2
(
q + q′)2
+1
4(1 + γ)2
(
q − q′)2}
= −ω04
{
1 + γ
1 − γ(
q − q′)2
+1 − γ1 + γ
(
q + q′)2}
.
(2.30)
So Eq. (2.28) can be rewritten in the following convenient form
〈
q′∣
∣ exp (−itH)∣
∣q〉
=
[
ω0
π(1 − γ2)
] 12
exp
{
−ω04
[
1 + γ
1 − γ(
q′ − q)2
+1 − γ1 + γ
(
q′ + q)2]}
.
(2.31)
This equation can be given a physical interpretation. Let’s define
AM(t) =
[
ω0
π(1 − γ2)
] 12
=
[
ω0
π(1 − exp(−2iω0t))
] 12
, (2.32)
and the following variable with the dimension of time
ϑM ≡2
iω0
1 − γ1 + γ
=2
iω0
1 − exp (−iω0t)1 + exp (−iω0t)
=2
ω0tan
(
1
2ω0t
)
. (2.33)
-
2.3 Temporal evolution in position representation 27
Through the above definitions, Eq. (2.31) can be written as
〈
q′∣
∣ exp (−itH)∣
∣q〉
= AM(t) exp
{
iϑM
[
1
2
(
q′ − qϑM
)2
− 12ω20
(
q′ + q2
)2]}
. (2.34)
In this equation, the term 12
(
q′−qϑM
)2can be interpreted as the classical kinetic energy, with
mean velocity
v =q′ − qϑM
, (2.35)
where ϑM is the time. The term12ω
20
(
q′+q2
)2represents the classical potential energy in
the mean position between the initial point q and the final point q′. At this point threeobservations are in order:
Observation 1: the quantity between [. . .] in Eq. (2.34) would represent the classical
lagrangian L = Ecin − Epot.
Observation 2: ϑM is not the time t but a periodical function of t (with period2πω0
),
as it is clear from its previous definition.
Observation 3: The transition amplitude of Eq. (2.34) must be solution of the Schrödinger
equation(
−i∂t −1
2
d2
dq′2+
1
2ω20q
′2 − 12ω0
)
〈
q′∣
∣ exp (−itH)∣
∣q〉
= 0 , (2.36)
with the initial condition
limt→0〈
q′∣
∣ exp (−itH)∣
∣q〉
=〈
q′∣
∣q〉 = δ(q′ − q) . (2.37)This condition is incompatible with the expression of Eq. (2.34), unless we adopt a math-
ematical regularization which introduces a negative imaginary part to the time t. In fact,
for t → 0, Eq. (2.34) we have〈
q′∣
∣ exp (−itH)∣
∣q〉→ 1√
2π itei(q′−q)2
2t . (2.38)
2.3.1 Infinitesimal time evolution
Let’s now specify Eq. (2.34) for an infinitesimal time interval ∆t. From the definition of
ϑM, Eq. (2.33), we have
∆ϑM ≃ ∆t . (2.39)The limit of AM(∆t) for small ∆t becomes:
AM(∆t) =
[
ω0
π(1 − exp(−2iω0t))
] 12
≃ 1√2π i∆t
=
(
1√2π∆t
exp(−iπ4
)
(∆t > 0)1√
2π∆texp
(
iπ4)
(∆t < 0)
)
(2.40)
-
28 2. Feynman path-integral quantization in quantum mechanics
Observe that AM(∆t) diverges in the limit ∆t → 0. The velocity of Eq. (2.35), in the limit∆t → 0 becomes
v =q′ − q∆t
= q̇ . (2.41)
As a consequence, the terms in square brackets in Eq. (2.34) represent the Lagrangian:
L(q, q̇) =1
2
(
q′ − q∆t
)2
− 12ω20
(
q′ + q2
)2
(2.42)
and we have〈
q′∣
∣ exp (−i∆tH)∣
∣q〉
≡ AM(∆t)ei∆tL(q,q̇) . (2.43)Observation: the exponent of Eq. (2.43) does not coincide with the power series expansion
of ∆t of the exponent of Eq. (2.34).
2.3.2 Path-integral
Let’s consider the evolution of the system from the initial state∣
∣q0〉
to the final state∣
∣qN〉
in a finite time T. We can divide T in N steps ∆t, all equal in size,
T = N∆t , (2.44)
with N large so that we can apply the equations of the previous section. We have
〈qN | exp (−iTH)∣
∣q0〉
= 〈qN |e−i ∑∆tH∣
∣q0〉
= 〈qN |e−i∆tH Ie−i∆tH I . . . Ie−i∆tH∣
∣q0〉
,
(2.45)
where we have introduced N − 1 Identity operators. Then, by using the completenessrelation of the position eigenstates
I =∫ +∞
−∞dq∣
∣q〉〈
q∣
∣ , (2.46)
and Eq. (2.43), we get
〈qN |e(−iTH)∣
∣q0〉
= [AM(∆t)]N
N−1∏i=1
∫ +∞
−∞dqi e
i∆tL(qN−1,q̇N−1) . . . ei∆tL(q0,q̇0)
= [AM(∆t)]N
N−1∏i=1
∫ +∞
−∞dqi e
i∆t ∑N−1j=0 L(q j,q̇ j)
≃ [AM(∆t)]NN−1∏i=1
∫ +∞
−∞dqi e
{iS(q0,...,qi,...,qN)} ,
(2.47)
-
2.4 Mathematical difficulties 29
where we have introduced the Action S, approximated by a discrete sum:
S(q0 , . . . , qi, . . . , qN) ≃ ∆tN−1∑j=0
L(q j , q̇ j) . (2.48)
The Action of the above equation is an ordinary function of the N + 1 variables q0, . . . , qN ,
which define, for discrete time intervals ∆t, the classical path between q0 and qN during
the time T. In Eq. (2.47) the endpoints q0 and qN are fixed, while on all intermediate
N − 1 qi positions we integrate. The probability amplitude for the evolution of thesystem from the initial state
∣
∣q0〉
to the final state∣
∣qN〉
is proportional to a weighted
sum over all classical paths kinematically allowed, where each path is weighted with
iS (with S being the classical action corresponding to the path). The larger is N, the
number of intermediate steps, the better is the approximation of the classical Action with
the discrete sum of Eq. (2.48). However it has to be noted that the limit N → ∞ (∆t → 0)in Eq. (2.47) is not possible because AM(∆t) diverges for ∆t → 0 (cfr. Eq. 2.40)). Bypassingfor the time being this problem, Eq. (2.47) is formally written as
〈qN |e(−iTH)∣
∣q0〉
≃ [AM(∆t)]N∫
[dq]eiS(q) , (2.49)
where the integration is understood as functional integration over the classical paths
We stress again that the knowledge of 〈q′|e(−iTH)∣
∣q0〉
is equivalent to the knowledge
of the Schrödinger equation. In fact the wave function at q’ is ψ(q′, t) ≡ 〈q′|ψ(t)〉. But∣
∣ψ(t)〉
= exp (−itH)∣
∣ψ(0)〉
. By projecting on the bra〈
q′∣
∣ both members of the above
equation and inserting the completeness relation of Eq. (2.46), we have
ψ(q′ , t) =∫
dq 〈q′|e(−iTH)|q〉|ψ(q, 0)〉 . (2.50)
The quantity 〈q′|e(−iTH)|q〉 is called the propagator.
2.4 Mathematical difficulties
If we consider Eq. (2.43) and try to obtain the initial condition of Eq. (2.37),〈
q′∣
∣q〉 =δ(q′ − q), we get
lim∆t→0〈
q′∣
∣e−i∆tH∣
∣q〉 ≃ lim∆t→0
1√2π i∆t
ei2(q′−q)2∆t 6= δ(q′ − q) , (2.51)
due to the presence of the imaginary unit i, unless we introduce a mathematical regular-
ization. If ∆t has a small imaginary part the initial condition is recovered.
Another problem is related to the group properties of the temporal translation (implicitely
used in Eq. (2.45)). To illustrate this point, we can rewrite Eq. (2.43) in the following way:
〈
q′∣
∣ exp (−i∆tH)∣
∣q〉
≡ AM(∆t)e[i2(
1∆t− 14ω20∆t)(q2+q′2)−i( 1∆t+ 14ω20∆t)q′q] . (2.52)
-
30 2. Feynman path-integral quantization in quantum mechanics
The problem consists in reproducing the following identity
〈
q′∣
∣e(−2i∆tH)∣
∣q〉
=∫ +∞
−∞dq
′′ 〈q′∣
∣e(−i∆tH)∣
∣q′′〉〈
q′′∣∣e(−i∆tH)
∣
∣q〉
(2.53)
trhough Eq. (2.52). In fact, using Eq. (2.52), we get for the right hand side of Eq. (2.53)
∫ +∞
−∞dq
′′ 〈q′∣
∣e(−i∆tH)∣
∣q′′〉〈
q′′∣∣e(−i∆tH)
∣
∣q〉
= [AM(∆t)]2 exp
[
i
2
(
1
∆t− 1
4ω20∆t
)
(
q2 + q′2)
]
∫ +∞
−∞dq
′′exp
[
i
(
1
∆t− 1
4ω20∆t
)
q′′2 − i
(
1
∆t+
1
4ω20∆t
)
(
q′ + q)
q′′]
.
(2.54)
The integral of the above equation is not defined because the coefficient of q′′2 is purely
imaginary. Also in this case a methematical regularization is required (if ∆t has a negative
imaginary part the integration converges).
2.5 Euclidean Time
We have seen in the previous section that the mathematical difficulties can be avoided
by giving the time a negative imaginary part. Usually a trick is to apply the following
transformation to the time
t → −iτ , (2.55)with τ real, which is nothing else than the Wick rotation you have already encountered
during the QED course. The time τ is the Euclidean time. At the end of the calculations
the physical results are obtained through analytic continuation (inverse Wick rotation),
which allows to recover the real time t.
With euclidean time, Eq. (2.34) becomes
〈
q′∣
∣ exp (−τH)∣
∣q〉
= AE(τ) exp
{
−ϑE[
1
2
(
q′ − qϑE
)2
+1
2ω20
(
q′ + q2
)2]}
, (2.56)
where
AE(τ) =
[
ω0
π(1 − γ2E)
]12
=
[
ω0
π(1 − exp(−2ω0τ))
]12
,
and
γE = e−ω0τ . (2.57)
The variable ϑE is defined in analogy with Eq. (2.33)
ϑE =2
ω0
1 − γ1 + γ
=2
ω0
1 − exp (−ω0τ)1 + exp (−ω0τ)
=2
ω0tanh
(
1
2ω0τ
)
(2.58)
-
2.5 Euclidean Time 31
and is not a periodical function of τ .
Observation: In analogy with Eq. (2.36), the transition amplitude of Eq. (2.56) satisfies
the following equation
(
∂τ −1
2
d2
dq′2+
1
2ω20q
′2 − 12ω0
)
〈
q′∣
∣ exp (−τH)∣
∣q〉
= 0 , (2.59)
with the initial condition
limτ→0〈
q′∣
∣ exp (−τH)∣
∣q〉
=〈
q′∣
∣q〉 = δ(q′ − q) . (2.60)
If we consider infinitesimal time steps, we have
∆ϑE ≃ ∆τ (2.61)
and
AE(∆τ) ≃1√
2π∆τ, (2.62)
which is still divergent for ∆τ → 0.The analogous of Eq. (2.43) for euclidean time is
〈
q′∣
∣ exp (−∆τH)∣
∣q〉
= AE(∆τ) exp
{
−∆τ[
1
2
(
q′ − q∆τ
)2
+1
2ω20
(
q′ + q2
)2]}
≡ AE(∆τ)e−∆τLE(q,q̇) ,(2.63)
where the Euclidean Lagrangian is
LE(q, q̇) ≡1
2
(
q′ − q∆τ
)2
+1
2ω20
(
q′ + q2
)2
. (2.64)
Notice that LE is positive definite.
With the euclidean time, the initial condition of Eq. (2.37) is recovered. In fact
lim∆τ→0〈
q′∣
∣ exp (−∆τH)∣
∣q〉
= lim∆τ→01√
2π∆τe−
12∆τ (q
′−q)2 = δ(q′ − q) . (2.65)
Also the second mathematical problem raised in the previous section is solved with
the adoption of the Euclidean Time. In fact, instead of Eq. (2.52), we get
〈
q′∣
∣ exp (−∆τH)∣
∣q〉
≡ AE(∆τ)e[
− 12( 1∆τ+ 14ω20∆τ)(q2+q′2)2+( 1∆τ− 14ω20∆τ)q′q
]
. (2.66)
and Eq. (2.53) becomes
〈
q′∣
∣e(−2∆τH)∣
∣q〉
=∫ +∞
−∞dq
′′ 〈q′∣
∣e(−∆τH)∣
∣q′′〉〈
q′′∣∣e(−∆τH)
∣
∣q〉
. (2.67)
-
32 2. Feynman path-integral quantization in quantum mechanics
From the above equations we have
∫ +∞
−∞dq
′′ 〈q′∣
∣e(−∆τH)∣
∣q′′〉〈
q′′∣∣e(−∆τH)
∣
∣q〉
= [AE(∆τ)]2 exp
[
−12
(
1
∆τ+
1
4ω20∆τ
)
(
q2 + q′2)2]
∫ +∞
−∞dq
′′exp
[
−(
1
∆τ+
1
4ω20∆τ
)
q′′2 +
(
1
∆τ− 1
4ω20∆τ
)
(
q′ + q)
q′′]
.
(2.68)
In this case the integral is well defined. The path-integral of Eq. (2.47), with Euclidean
Time, becomes
〈qN |e(−TH)∣
∣q0〉
= [AE(∆τ)]N
N−1∏i=1
∫ +∞
−∞dqi e
−∆τ ∑N−1j=0 LE(q j ,q̇ j)
≃ [AE(∆τ)]NN−1∏i=1
∫ +∞
−∞dqi e
{−SE(q0,...,qi,...,qN)} ,(2.69)
where the Euclidean Action SE is related to the Euclidean Lagrangian as follows:
SE(q0, . . . , qN) ≃ ∆τN−1∑j=0
LE(q j, q̇ j) . (2.70)
Eq. (2.69) gives a positive measure to the classical paths, contrary to Eq. (2.47).
However, the divergence of AE(∆τ) for∆τ → 0 is not solved by going from the Minkowskianto the Euclidean Time. The usual way of writing Eq. (2.69) is
〈qN |e(−TH)∣
∣q0〉 ≃ [AE(∆τ)]N
∫
[dq] e−SE[q] . (2.71)
2.5.1 The classical limit and the semiclassical approximation
According to Eq. (2.71) we can recover the classical dynamics. In fact the path for which
the Action is minimized is the one that maximizes the integral.
To see this more directly, even if in a heuristical way, let us reinsert explicitely the Planck
constant and use Minkowskian time in Eq. (2.49)
〈qN |e(−ih̄ TH)
∣
∣q0〉
≃ [AM(∆t)]N∫
[dq]eih̄ S(q) . (2.72)
The classical limit is defined as the limit of the propagator for h̄ → 0. By inspectionof Eq. (2.72), we see that the limit is fullfilled when S(q) ≫ h̄. In fact suppose that atrajectory qc(t) exists, such that qc(t0) = q0 and qc(t0 + T) = qN , which extremizes the
Action. The condition δS = 0 implies that trajectories close to qc(t) contribute to the
-
2.5 Euclidean Time 33
integral of Eq. (2.72) with equal or very similar phases, thus interfering in constructive
way. On the contrary, close to any trajectory which does not extremize the Action we will
find other trajectories with very different phases and therefore interfering in destructive
way. So the main contribution to the integral comes from a set of trajectories close to qc(t),
where “close” means that the related Action differs at most from S(qc) by about h̄. This
means that in the limit h̄ → 0 the motion of the system is well described by the classicaltrajectory. Moreover, the principle of least Action of classical mechanics can be thought
of as a particular limit of quantum mechanics.
According to the above argument, quantum mechanics describes the fluctuations of the
Action in a narrow range around the classical path. Thus we can expand the Action
functional in terms of fluctuations around the classical path qc(t):
S[q, q̇] = Scl +1
2
∫
(
δ2L
δq2(δq)2 + 2
δ2L
δqδq̇δqδq̇ +
δ2L
δq̇2(δq̇)2
)
+ . . . ≡ Scl +1
2δ2S + . . . ,
(2.73)
where δq is the fluctuation around the classical path. The derivatives have to be taken at
the classical path. Since the Action is stationary at the classical path, Eq. (2.73) does not
include first derivative terms. The propagator of Eq. (2.72) becomes
〈qN |e(−ih̄ TH)
∣
∣q0〉
≃ [AM(∆t)]N eih̄ Scl
∫
[dq] ei
2h̄δ2S + . . . . (2.74)
The approximation of calculating the propagator according to Eq. (2.74) is called semiclas-
sical approximation. If the Lagrangian depends at most quadratically on q and q̇, Eq. (2.74)
is exact, without higher-order terms.
2.5.2 The free particle propagator
With the results of the previous sections, we can easily obtain the propagator for a free
particle of mass m moving in one dimension. In fact we can refer to the Hamiltonian of
Eq. (2.3), introducing the mass m and taking the limitω0 → 0. From Eqs.(2.33) and (2.32),we get
limω0→0ϑ = t
AM(t) =
√
m
2iπh̄t.
(2.75)
The analogous of Eq. (2.34) is
K(t; q′ , q) ≡ 〈q′∣
∣ exp
(
− ih̄
tH0
)
∣
∣q〉
=
√
m
2iπh̄texp
{
im (q′ − q)2
2h̄t
}
. (2.76)
The phase factor in the square root should be taken as in Eq. (2.40), according to the
inverse Wick rotation, starting from the euclidean propagator
KE(τ ; q′, q) =
√
m
2πh̄τexp
{
−m (q′ − q)2
2h̄τ
}
. (2.77)
-
34 2. Feynman path-integral quantization in quantum mechanics
The generalization to three dimensions is easily obtained as
K(t;~r′ ,~r) =≡〈
~r′∣
∣ exp
(
− ih̄
tH0
)
∣
∣~r〉
=( m
2iπh̄t
)32
exp
{
+im |~r′ −~r|2
2h̄t
}
. (2.78)
The free particle propagator in momentum representation
Since for a free particle the momentum is conserved, it is instructive to take the Fourier
transform of Eq. (2.76)
K̃(p, t) =∫ +∞
−∞d(∆q) exp
{
−i ph̄∆q}
K(∆q, t)
=
√
m
2iπh̄t
∫ +∞
−∞d(∆q) exp
{
−i ph̄∆q}
exp
{
im(∆q)2
2h̄t
}
. (2.79)
The equivalent expression with euclidean time is
K̃E(p, τ) =
√
m
2πh̄τ
∫ +∞
−∞d(∆q) exp
{
−i ph̄∆q}
exp
{
−m(∆q)2
2h̄τ
}
. (2.80)
The integration of Eq. (2.80) can be performed by Eq. (2.202), with a = m2h̄τ and b = −iph̄ ,
obtaining:
K̃E(p, t) = exp
{
−1h̄
p2
2mτ
}
(2.81)
and 1
K̃(p, t) = exp
{
− ih̄
p2
2mt
}
. (2.83)
We can now make a Fourier antitranform
K(∆q, t) =1
2πh̄
∫ +∞
−∞dp exp
{
+ip
h̄∆q}
˜K(p, t)
=1
2πh̄
∫ +∞
−∞dp exp
{
+ip
h̄∆q}
exp
{
− ih̄
p2
2mt
}
=1
2πh̄
∫ +∞
−∞dp exp
{
+i
h̄
(
p∆q − p2
2mt
)}
. (2.84)
As discussed in Section 2.2, the boundary condition K = 0 for t < 0 is understood. We
can implement it multiplying the integrand of Eq. (2.84) by the step function ϑ(t) and
1The same result of Eq. (2.83) can be obtained starting from the definition
K(p, p′, t) =〈
p′∣
∣ exp
{
− ih̄
tH0
}
∣
∣p〉
=〈
p′∣
∣ exp
{
− ih̄
t( p̂)2
2m
}
∣
∣p〉
= exp
{
− ih̄
p2
2mt
}
δ(p − p′) . (2.82)
-
2.5 Euclidean Time 35
recalling the integral representation (which can be verified through application of the
residue theorem)
ϑ(t) =1
2π i
∫ +∞
−∞dω
eiωt
ω− iε with ε > 0 : (2.85)
K(∆q, t) =h̄
i
∫ +∞
−∞dp
2πh̄
dω
2πh̄
exp{
+ ih̄
[
p∆q −(
p2
2m − h̄ω)
t]}
ω− iε . (2.86)
The coefficient of t in the plane wave is the energy, so we can make the substitution
E =p2
2m− h̄ω , (2.87)
obtaining for Eq. (2.86)
K(∆q, t) =h̄
i
∫ +∞
−∞dp
2πh̄
dE
2πh̄
exp{
+ ih̄
[
p∆q −(
p2
2m − E)
t]}
E − p22m + iε. (2.88)
Eq. (2.88) says that the propagation takes place also for energies E 6= p22m . The classicaldispersion relation E = p
2
2m is the pole of the propagator.
2.5.3 Particle in one dimension with generic potential
In this section we give the treatment for a generic one dimensional quantum system, of
mass m, described by the coordinate q, the conjugate momentum p and the Hamiltonian
H(q̂, p̂) = K ( p̂) + V (q̂) = p̂2
2m+ V (q̂) . (2.89)
Let us recall the transition amplitude from the initial state∣
∣q0〉
to the final state∣
∣qN〉
during the finite time T, as in Eq. (2.45), which we report here
〈qN | exp (−i
h̄TH)
∣
∣q0〉
= 〈qN |e−ih̄ ∑∆tH
∣
∣q0〉
= 〈qN |e−ih̄∆tH Ie−
ih̄∆tH I . . . Ie−
ih̄∆tH
∣
∣q0〉
,
and we replace the identity operators I with the completeness relation of the position
eigenstates of Eq. (2.46)
I =∫ +∞
−∞dq∣
∣q〉〈
q∣
∣ .
-
36 2. Feynman path-integral quantization in quantum mechanics
Since the kinetic part K and the potential V of the Hamiltonian don’t commute, we have,
e−ih̄∆t(V+K) = 1 − i
h̄∆t(K + V)− (∆t)
2
2h̄2
(
K2 + V2 + VK + KV)
+O[(∆t)3]
e−ih̄∆tVe−
ih̄∆tK = 1 − i
h̄∆t(K + V)− (∆t)
2
2h̄2
(
K2 + V2 + 2VK)
+O[(∆t)3 ] ,(2.90)
so that we can write,
e−ih̄∆tH = e−
ih̄∆tVe−
ih̄∆tK +O[(∆t)2 ] . (2.91)
An error of order (∆t)2, iterated N times, gives a globar error of order ∆t, which can be
neglected when we take the limit of small ∆t. Eq. (2.45) can be written as
〈qN | exp (−i
h̄TH)
∣
∣q0〉
=
[
N−1∏i=1
∫ +∞
−∞dqi
]
〈qN |e−ih̄∆tH
∣
∣qN−1〉
. . . 〈q1|e−ih̄∆tH
∣
∣q0〉
≃[
N−1∏i=1
∫ +∞
−∞dqi
]
〈qN |e−ih̄∆tVe−
ih̄∆tK
∣
∣qN−1〉
. . . 〈q1|e−ih̄∆tVe−
ih̄∆tK
∣
∣q0〉
.
(2.92)
Now let us consider a single transition amplitude
〈q j+1|e−ih̄∆tV(q̂)e−
ih̄∆tK( p̂)
∣
∣q j〉
= e−ih̄∆tV(q j+1)〈q j+1|e−
ih̄∆tK( p̂)
∣
∣q j〉
. (2.93)
Now, remembering that
〈q∣
∣p〉
=1√2πh̄
eih̄ pq
∫
dp∣
∣p〉〈
p∣
∣ = I
〈p′∣
∣p〉
= δ(p′ − p) ,
(2.94)
we have
〈q j+1|e−ih̄∆tK( p̂)
∣
∣q j〉
=∫ dp′j√
2πh̄
∫ dp j√2πh̄
eih̄
[
p′jq j+1−p jq j]
e−ih̄∆tK(p j)δ(p j − p′j)
=∫ dp j
2πh̄e
ih̄ [(p jq j+1−p jq j)−∆tK(p j)] ,
(2.95)
where p j is the momentum between q j and q j+1. By means of the above result, Eq. (2.93)
can be written as
〈
q j+1∣
∣e−ih̄∆tV(q̂)e−
ih̄∆tK( p̂)
∣
∣q j〉
=1
2πh̄
∫
dp j eih̄ [p j(q j+1−q j)−∆t(K(p j)+V(q̄ j))]
=1
2πh̄
∫
dp j eih̄ [p j(q j+1−q j)−∆tH(p j ,q̄ j)] ,
(2.96)
-
2.5 Euclidean Time 37
where we have defined q̄ j =12
(
q j + q j+1)
. Inserting Eq. (2.96) in Eq. (2.92), we get
〈qN | exp(
− ih̄
TH
)
∣
∣q0〉
=
[
N−1∏i=1
∫
dqi
] [
N−1∏j=0
∫ dp j
2πh̄
]
exp
{
i
h̄
N−1∑i=1
[pi (qi+1 − qi)− ∆tH (pi, q̄i)]}
.
(2.97)
Eq. (2.97) is the defining (discretized) equation for the “functional integral” (in the con-
tinuum limit)
〈qN | exp(
− ih̄
TH
)
∣
∣q0〉
=∫
[dq][dp]
2πh̄exp
{
i
h̄
∫ tN
t0dt [pq̇ − H(p, q)]
}
. (2.98)
If the kinetic term is of the form K(p̂) =p̂2
2m and V(q) is a quadratic polinomial, then the
integration over pi in Eq. (2.97) can be performed analytically by completing the square
in the argument of the exponential, obtaining
〈qN | exp(
− ih̄
TH
)
∣
∣q0〉
=[ m
2πh∆t
]N2
[
N−1∏i=1
∫
dqi
]
exp
{
i∆t
h̄
N−1∑i=0
[
m
2
(
qi+1 − qi∆t
)2
− V (q̄i)]}
.
(2.99)
Eq. (2.99) is the defining (discretized) equation for the “functional integral” (in the con-
tinuum limit)
〈qN | exp(
− ih̄
TH
)
∣
∣q0〉
= N∫
[dq] exp
{
i
h̄
∫ tN
t0dt L (q, q̇)
}
, (2.100)
where N is the constant (already seen in previous sections) which becomes ill defined inthe continuum limit. The integral
∫
L(q, q̇)dt =∫
(
mq̇2
2 − V(q))
dt at the exponent gives
the classical action along the given path.
Eq. (2.100) was adopted by Feynman as the starting point to derive the Schrödinger equa-
tion. However, the most general expression is Eq. (2.98), while Eq. (2.100) relyes on the
form of the Hamiltionian H(p̂, q̂) =p̂2
2m + V(q). If, for instance, we have a velocity-
dependent potential, the Lagrangian at in Eq. (2.100) needs to be modified to get the
correct result. Something similar happens with non-Abelian gauge theories.
Observation: the propagator is written at the beginning in terms of non-commuting oper-
ators q̂ and p̂, while in the end the functional integral is expressed in terms of c–numbers
only, so that ambiguities can originate if we have products of q̂ and p̂ in the Hamiltonian.
This ambiguities are not present if we start from a “Weyl ordered” Hamiltonian, where
the q̂ operators appear symmetrically at the left and right of p̂. For instance
〈
qk+1∣
∣
1
4
(
q̂2 p̂2 + 2q̂ p̂2q̂ + p̂2q̂2)
∣
∣qk〉
=
(
qk+1 + qk2
)2〈
qk+1∣
∣ p̂2∣
∣qk〉
, (2.101)
without ambiguity.
-
38 2. Feynman path-integral quantization in quantum mechanics
Observation: the time integral in Eq. (2.100) does not converge without a regularization:
the integrand e(i/h̄)S has modulus equal 1. A possible solution is to adopt Euclidean time
t = −iτ and perform an inverse analytic continuation at the end, which is the solutionadopted in the previous sections. Here we illustrate the alternative possibility of giving a
small negative imaginary part to the time: t = (1 − iχ)τ
dt = (1 − iχ)dτ
q̇ =dq
dt= (1 + iχ)
dq
dτ
(2.102)
With this transformation, the integrand Eq. (2.100) becomes
eih̄ Sχ = exp
{
i
h̄
∫
dτ
[
m
2
(
dq
dτ
)2
− V(q)]}
· exp{
−χ∫
dτ
[
m
2
(
dq
dτ
)2
+ V(q)
]}
.
(2.103)
The integrand exp(
ih̄ Sχ
)
has modulus equal to exp (−χI), where
I =∫
dτ
[
m
2
(
dq
dτ
)2
+ V(q)
]
=∫
dτ H(q, q̇) . (2.104)
Concerning the convergence of the functional integral, we can distinguish the following
cases:
• V(q) = 0 this is the free particle, which we have already treated in detail; the func-tional integral is convergent.
• V(q) positively defined: I > I0, where I0 is the value calculated with the same tra-jectory for V(q) = 0. The degree of convergence is the same as for V(q) = 0.
• V(q) bounded from below: V(q) > V0. In this case I > I0 + V0T. Adding theconstant V0T does not change the convergence w.r.t. the two previous cases.
• V(q) not bounded from below: there is no general rule. For instance, if V(q) = −qn,the convergence of the functional integral depends on the value of n. We only quote
in passing that the integral converges if −1 ≤ n ≤ 0 and does not converge ifn < −1 or n > 0. The Coulomb potential is a limiting case.
The cases not covered by the above classes fail also with other formulations of quantum
mechanics.
2.5.4 Periodical paths in Euclidean Time
Consider the limit of the following trace:
limT→∞∞
∑n=0
〈n|e−2TH |n〉 = limT→∞∞
∑n=0
(
e−2Tω0)n
= limT→∞1
1 − e−2Tω0 = 1 . (2.105)
-
2.6 Feynman path integral for Euclidean Green Fuctions 39
Since the trace is independent of the representation, we can compute it by means of the
basis of position eigenstate∣
∣q〉
:
limT→∞∫
∞
−∞dq 〈q|e−2TH |q〉 = 1 . (2.106)
We can now express the above equation by means of the path-integral in the following
way:
1 = limT→∞∫
∞
−∞dq 〈q|e−2TH |q〉
= limT→∞∫
∞
∞
dq 〈q|e−∆TH Ie−∆TH . . . Ie−∆TH|q〉
= limT→∞∫
∞
−∞dq−N+1 . . .
∫
∞
−∞dqN〈qN |e−∆TH|qN−1〉〈qN−1| . . . |q−N+1〉〈q−N+1|e−∆TH|q−N = qN〉
= limT→∞ [AE(∆τ)]2N∫ +∞
−∞dq−N+1 . . .
∫
∞
−∞dqNe
−∆τLE(qN−1,q̇N−1) . . . e−∆τLE(q−N ,q̇−N)
= limT→∞ [AE(∆τ)]2N∫ +∞
−∞dq−N+1 . . .
∫
∞
−∞dqNe
−SE(q−N=qN ,...,qN) .
(2.107)
Observation 1: because of the trace operation, the classical paths over which we have to
sum are periodic in the euclidean time 2T.
Observation 2: because of the term [AE(∆τ)]2N the limit ∆T → 0 can not be taken.
We will see in the following section how to use Eq. (2.107) and circumvent the problem.
2.6 Feynman path integral for Euclidean Green Fuctions
We recall that a Green Function is defined as the vacuum expectation value of a time
ordered product of field operators:
G(t1 , . . . , tn) ≡ 〈0|T[q̂H(t1) . . . q̂H(tn)]∣
∣0〉
. (2.108)
We will start from Euclidean Green Functions and express them in terms of the Feynman
path-integral. Let’s define the Heisenberg description for euclidean time (in analogy with
the Minkowskian case):
q̂H(τ) = eτH q̂se
−τH , (2.109)
where q̂s is the position operator in Schrödinger description. By observing that
limT→∞∞
∑n=0
〈n|e−TH q̂H(τ1)q̂H(τ2)e−TH∣
∣n〉
= limT→∞∞
∑n=0
e−2nTω0〈n|q̂H(τ1)q̂H(τ2)∣
∣n〉
= 〈0|q̂H(τ1)q̂H(τ2)∣
∣0〉
,
(2.110)
-
40 2. Feynman path-integral quantization in quantum mechanics
we can express the two points euclidean Green function as follows:
G(2)E (τ1, τ2) ≡ 〈0|T [q̂H(τ1)q̂H(τ2)]
∣
∣0〉
= limT→∞∞
∑n=0
〈n|e−THT [q̂H(τ1)q̂H(τ2)] e−TH∣
∣n〉
= limT→∞∫ +∞
−∞dq 〈q|e−THT [q̂H(τ1)q̂H(τ2)] e−TH
∣
∣q〉
= ϑ(τ1 − τ2)limT→∞∫ +∞
−∞dq 〈q|e−TH q̂H(τ1)q̂H(τ2)e−TH
∣
∣q〉
+ ϑ(τ2 − τ1)limT→∞∫ +∞
−∞dq 〈q|e−TH q̂H(τ2)q̂H(τ1)e−TH
∣
∣q〉
.
(2.111)
The trace of the above two matrix elements can be performed by means of the Feynman
path integral. Taking the time interval 2T as in Eq. (2.107) divided in 2N equal su