Path Integrals in Physics
Vo l um e I I
Quantum Field Theory, Statistical Physics and
other Modern Applications

Path Integrals in Physics
Volume II
Quantum Field Theory, Statistical Physics
and other Modern Applications
M Chaichian
Department of Physics, University of Helsinki
and
Helsinki Institute of Physics, Finland
and
ADemichev
Institute of Nuclear Physics, Moscow State University, Russia
Institute of Physics Publishing
Bristol and Philadelphia
c
 IOP Publishing Ltd 2001
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise,
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Committee of Vice-Chancellors and Principals.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN 0 7503 0801 X (Vol. I)
0 7503 0802 8 (Vol. II)
0 7503 0713 7 (2 Vol. set)
Library of Congress Cataloging-in-Publication Data are available

3.2.5 Covariant generating functional in the Yang–Mills theory 67
3.2.6 Covariant perturbation theory for Yang–Mills models 73
3.2.7 Higher-order perturbation theory and a sketch of the renormalization procedure
for Yang–Mills theories 80
3.2.8 Spontaneous symmetry-breaking of gauge invariance and a brief look at the
standard model of particle interactions 88
3.2.9 Problems 98
3.3 Non-perturbative methods for the analysis of quantum field models in the path-integral
approach 101
3.3.1 Rearrangements and partial summations of perturbation expansions: the 1/N-
expansion and separate integration over high and low frequency modes 101
3.3.2 Semiclassical approximationin quantum field theory and extendedobjects (solitons)110
3.3.3 Semiclassical approximation and quantum tunneling (instantons) 120
3.3.4 Path-integral calculation of quantum anomalies 130
3.3.5 Path-integral solution of the polaron problem 137
3.3.6 Problems 144
3.4 Path integrals in the theory of gravitation, cosmology and string theory: advanced
applications of path integrals 149
vi
Contents
3.4.1 Path-integral quantization of a gravitational field in an asymptotically flat
spacetime and the corresponding perturbation theory 149
3.4.2 Path integrals in spatially homogeneous cosmological models 154
3.4.3 Path-integral calculation of the topology-changetransitions in (2+1)-dimensional
gravity 160
3.4.4 Hawking’s path-integral derivation of the partition function for black holes 166
3.4.5 Path integrals for relativistic point particles and in the string theory 174
3.4.6 Quantum field theory on non-commutative spacetimes and path integrals 185
4 Path integrals in statistical physics 194
4.1 Basic concepts of statistical physics 195

4.7.3 Problems 308
Supplements 311
I Finite-dimensional Gaussian integrals 311
II Table of some exactly solved Wiener path integrals 313
III Feynman rules 316
IV Short glossary of selected notions from the theory of Lie groups and algebras 316
Contents
vii
V Some basic facts about differential Riemann geometry 325
VI Supersymmetry in quantum mechanics 329
Bibliography 332
Index 337

Preface to volume II
In the second volume of this book (chapters 3 and 4) we proceed to discuss path-integral applications
for the study of systems with an infinite number of degrees of freedom. An appropriate description of
such systems requires the use of second quantization, and hence, field theoretical methods. The starting
point will be the quantum-mechanical phase-space path integrals studied in volume I, which we suitably
generalize for the quantization of field theories.
One of the central topicsof chapter 3 is the formulation of the celebrated Feynman diagram technique
for the perturbation expansion in the case of field theories with constraints (gauge-field theories),
which describe all the fundamental interactions in elementary particle physics. However, the important
applications of path integrals in quantum field theory go far beyond just a convenient derivation of the
perturbation theory rules. We shall consider, in this volume, various modern non-perturbativemethods for
calculations in field theory, such as variational methods, the description of topologically non-trivial field
configurations, the quantization of extended objects (solitons and instantons), the 1/N-expansion and the
calculation of quantum anomalies. In addition, the last section of chapter 3 contains elements of some
advanced and currently developing applications of the path-integral technique in the theory of quantum
gravity, cosmology, black holes and in string theory.
For a successful reading of the main part of chapter 3, it is helpful to have some acquaintance with

and discuss applications of the latter to systems with an infinite number of degrees of freedom. In other
words, we shall derive path-integral representations for different objects in quantum field theory (QFT).
Of course, this is nothing other than quantum mechanics for systems with an arbitrary or non-conserved
number of excitations (particles or quasiparticles). Therefore, the starting point for us is the quantum-
mechanical phase-space path integrals studied in chapter 2. In most practical applications in QFT, these
path integrals can be reduced to the Feynman path integrals over the corresponding configuration spaces
by integrating over momenta. This is especially important for relativistic theories where this transition
allows us to keep relativistic invariance of all expressions explicitly.
Apparently, the most important result of path-integral applications in QFT is the formulation of the
celebrated Feynman rules for perturbation expansion in QFT with constraints, i.e. in gauge-field theories
which describe all the fundamental interactions of elementary particles. In fact, Feynman derived his
important rules (Feynman 1948, 1950) (in quantum electrodynamics (QED)) just using the path-integral
approach! Later, these rules (graphically expressed in terms of Feynman diagrams) were rederived in
terms of the standard operator approach. But the appearanceof more complicated non-Abelian gauge-field
theories (which describe weak, strong and gravitational interactions) again brought much attention to the
path-integral method which had proved to be much more suitable in this case than the operator approach,
because the latter faces considerable combinatorial and other technical problems in the derivation of the
Feynman rules. In fact, it is this success that attracted wide attention to the path-integral formalism in
QFT and in quantum mechanics in general.
Further development of the path-integral formalism in QFT has led to results far beyond the
convenientderivation of perturbation theory rules. In particular, it has resulted in various non-perturbative
approximations for calculations in field theoretical models, variational methods, the description of
topologically non-trivial field configurations, the discovery of the so-called BRST (Becchi–Rouet–Stora–
Tyutin) symmetry in gauge QFT, clarification of the relation between quantization and the theory of
stochastic processes, the most natural formulationof string theory which is believed to be the most realistic
candidate for a ‘theory of everything’, etc.
In the first section of this chapter, we consider path-integral quantization of the simplest field theories,
including scalar and spinor fields. We derive the path-integral expression for the generating functional
of the Green functions and develop the perturbation theory for their calculation. In section 3.2, after
an introduction to the quantization of quantum-mechanical systems with constraints, we proceed to the

3.1.1 Systems with an infinite number of degrees of freedom and quantum field theory
There are various formulations of quantum field theory, differing in the form of presentation
of the basic quantities, namely transition amplitudes. In the operator approach, the transition
amplitudes are expressed as the vacuum expectation value of an appropriate product of particle
creation and annihilation operators. These operators obey certain commutation relations
(generalization of the standard canonical commutation relations to a system with an infinite
number of degrees of freedom). Another formulation is based on expressing the transition
amplitudes in terms of path integrals over the fields. In studying the gauge fields, the path-
integral formalism has proven to be the most convenient. However, for an easier understanding
of the subject we shall start by considering unconstrained fields and then proceed to gauge-field
theories (i.e. field theories with constraints).
Let us consider, as a starting example, a single scalar field. From the viewpoint of
Hamiltonian dynamics, a field is a system with an infinitely large number of degrees of freedom,
for the field is characterized by a generalized coordinate ϕ(x) and a generalized momentum π(x)
at each space point x ∈
d
.
It is worth making the following remark. If we were intending to provide an introduction to
the very subject of quantum field theory, it would be pedagogically more reasonable to start
from non-relativistic many-body problems and the corresponding non-relativistic quantum field
Path-integral formulation of the simplest quantum field theories
3
aaa
q
k−1
q
k−2
q
k
q

K

k=1
1
2
[p
2
k
+ 
2
(q
k
− q
k+1
)
2
+ 
2
0
q
2
k
] (3.1.1)
where p
k
, q
k
(k = 1, ,K) are the canonical variables (momentum and position) of the kth
oscillator and the equations of motion read:
˙q

) − 
2
0
q
k
. (3.1.3)
The frequency 
0
defines the potential energy of an oscillator due to a shift from its equilibrium
position and the frequency  defines the interaction of an oscillator with its neighbours. Since
we shall use this model as a starting point for the introduction of quantum fields, a concrete
4
Quantum field theory: the path-integral approach
value of the particle masses in (3.1.1) is not important and for convenience we have put it equal
to unity (cf (2.1.42)). Besides, as is usual in relativistic quantum field theory, we use units such
that
= 1.
The equations of motion must be accompanied by some boundary conditions. Since we
are going to pass later to systems in infinite volumes (of infinite sizes), the actual form of
the boundary conditions should not have a crucial influence on the behaviour of the systems.
Therefore, we can choose them freely and the most convenient one is the periodic condition:
q
k+K
= q
k
. (3.1.4)
After the quantization, the canonical variables become operators with the following
canonical commutation relations:
[q
k

k+1
)
2
+ 
2
0
q
2
k
] (3.1.6)
it is helpful to introduce new variables (the so-called
normal coordinates
)

Q
r
,

P
r
via the discrete
Fourier transform:
q
k
=
1
√
K
K/2


]=iδ
rs
[

Q
r
,

Q
s
]=[

P
r
,

P
s
]=0
(3.1.8)
where r and s are integers from the interval [−K/2 +1, K/2]. It is easy to verify that the normal
coordinates also satisfy the periodic conditions:

Q
−K/2
=

Q
K/2
and

k
=

P
−k
. (3.1.9)
The Kronecker symbol δ
ln
can be represented as the sum
K

k=1
e
i2πk(l−n)/K
= Kδ
ln
. (3.1.10)
Path-integral formulation of the simplest quantum field theories
5
This is an analog of the integral representation (1.1.22) for the δ-function, adapted to the
discrete finite lattice with a periodic boundary condition. Using this formula, we can invert the
transformation (3.1.7) of the dynamical variables:

Q
r
=
1
√
K
K

2
K/2

r=−K/2+1
[

P
r

P
†
r
+ ω
2
r

Q
r

Q
†
r
] (3.1.12)
ω
2
r
≡ 
2

2sin

)
a
†
r
=
1
√
ω
r
(ω
r

Q
†
r
− i

P
r
)
(3.1.14)
(note that a
−r
= a
†
r
). The commutation relations for a
r
, a
†

(a
†
r
a
r
+
1
2
). (3.1.16)
Eigenstates of the Hamiltonian written in the latter form can be constructed in the standard way:
the state
|n
−K/2+1
, n
−K/2+2
, ,n
K/2
=
K/2

r=−K/2+1
1
√
n
r
!
(a
†
r
)

Let us consider the continuous limit for a chain of coupled oscillators K →∞, a → 0,with
a finite value of the product aK ≡ L. Technically, this corresponds to the following substitutions:
q
k
−→
q(x)
√
a

k
−→
1
a

L
0
dx  −→
v
a
(3.1.21)
and Hamiltonian (3.1.1) takes the following form in the limit
H =

L
0
dx
1
2

p

i2πr/L
Q
r
p(x) =
1
√
L
∞

r=−∞
e
i2πr/L
P
r
(3.1.23)
though the index r is now an arbitrary unbounded integer. The quantum Hamiltonian can be
cast again into the form (3.1.12) or (3.1.16):

H =
1
2
∞

r=−∞
(

P
r

P

r
= v
2
k
2
+ 
2
0
k ≡
2πr
L
(3.1.25)
with the only difference begin that the sums run over all integers. The eigenstates and
eigenvalues of this Hamiltonian are given by (3.1.17)–(3.1.20). The essentially new feature
of this system with an
infinite
number of degrees of freedom (i.e. after the transition K →∞)is
that the energy (3.1.19) of the lowest eigenstate |0 becomes
infinite
. We can circumvent this
difficulty by redefining the Hamiltonian as follows:

H −→

H − E
0
=
1
2
∞

by vectors. However, if we assume that for some reason the displacements are confined to
one
direction, we obtain the physically important case of
scalar
quantum fields ˆϕ(x, t), ˆπ(x, t).
Hamiltonians for quantum fields in higher-dimensional spaces are the direct generalizations
of those for the one-dimensional case (cf (3.1.22)). In particular, for the most realistic three-
dimensional space, we have

H =
1
2

d
3
r [ˆπ
2
(r, t) +v
2
(∇ ˆϕ(r, t))
2
+ 
2
0
ˆϕ
2
(r, t)]. (3.1.28)
The operators of the quantum field ˆϕ(r, t) and the corresponding momentum ˆπ(r, t) satisfy the
canonical commutation relations at equal times:
[ˆϕ(r, t), ˆπ(r

)
−1/2
[e
i(k·r−ω
k
t)
a
k
+ e
−i(k·r−ω
k
t)
a
†
k
] (3.1.31)
k
x,y,z
=
2πl
x,y,z
L
(3.1.32)
ω
2
k
= v
2
k
2

relations:
[a
k
,a
†
k

]=δ
kk

[a
k
,a
k

]=[a
†
k
,a
†
k

]=0.
(3.1.35)
8
Quantum field theory: the path-integral approach
The eigenstates and eigenvalues of this Hamiltonian again have the form (3.1.17)–(3.1.20),
so that the energy of a state |n
k
, n

ω
k
i
. (3.1.36)
Note that, if we put
v = c 
0
= mc
2
(3.1.37)
where c is the speed of light and m is the mass of a particle, equation (3.1.33) exactly coincides
with the relativistic relation between energy, mass and momentum k of a particle. Hence,
expression (3.1.36) for the energy of the quantum field ϕ(x, t) can be interpreted as the sum
of energies of the set (defined by the occupation numbers {n
k
i
}) of free relativistic particles. To
simplify the formulae, we shall, in what follows, put the speed of light equal to unity, c = 1;the
latter can be achieved by an appropriate choice of units of measurement.
• Thus, we have obtained a remarkable result: a quantum field with the Hamiltonian (3.1.28)
(or (3.1.22)) and the choice of parameters as in (3.1.37) is equivalent to a system of
an arbitrary number of free relativistic particles. According to the commutation relations
(3.1.35), these particles obey Bose–Einstein statistics.
We have already mentioned the specific problems of quantum systems with an infinite
number of degrees of freedom, that is, the appearance of divergent expressions. One example
is the energy of ‘zero oscillations’ (3.1.19), which diverges for an infinite number of oscillators.
Another example is the expression for ‘zero fluctuations’ of the field ϕ(t, r), in other words for the
dispersion of the field in the lowest energy state:
(
|0

2
→∞. (3.1.38)
The reason for the infinite value of the fluctuation is related to the fact that ˆϕ,actingonan
arbitrary state with finite energy, gives a state with an infinite norm. Thus ˆϕ does not belong
to well-defined operators in the Hilbert space of states of the Hamiltonian under consideration.
Another way to express this fact is to say that ˆϕ is an operator-valued distribution (generalized
function). To construct a well-defined operator, we have to smear ˆϕ with an appropriate test
function, e.g., to consider the quantity
¯ϕ
λ
def
≡
1
(2πλ
2
)
3/2

d
3
r e
−r
2
/(2λ
2
)
ˆϕ(t, r) (3.1.39)
which can be interpreted as an average value of the field in the volume λ
3
around the point r.


d
3
r π(r, t) ˙ϕ(r, t) − H[π(r, t), ϕ(r, t)]. (3.1.41)
The momentum π on the right-hand side of (3.1.41) is assumed to be expressed through ˙ϕ, ϕ
with the help of the Hamiltonian equation of motion. In our case,
˙ϕ ={ϕ, H}=π (3.1.42)
(recall that {·, ·} is the Poisson bracket). Thus, the Lagrangian for the scalar field reads as
L(t) =

d
3
r
1
2
[˙ϕ
2
(r, t) −(∇ϕ(r, t))
2
− m
2
ϕ
2
(r, t)]. (3.1.43)
To demonstrate the invariance of the Lagrangian (3.1.43) with respect to transformations forming
relativistic kinematic groups, i.e. the
Lorentz
or
Poincar
´

g
µν
y
ν
(repeating indices are assumed to be summed over). In particular, the squared vector in the
Minkowski space reads as
x
2
≡ (x
µ
)
2
= x
µ
g
µν
x
ν
= (x
0
)
2
− (x
1
)
2
− (x
2
)
2

µν
dx
ν
= (dt)
2
− (dr)
2
. (3.1.47)
In the literature on relativistic field theory, it is common to drop boldface type for four-dimensional
vectors and we shall follow this custom. If the vector indices µ,ν, take, in some expressions,
10
Quantum field theory: the path-integral approach
only spacelike values 1, 2, 3, we shall denote them by Latin letters l, k, and use the following
shorthand notation:
A
l
B
l
=
3

l=1
A
l
B
l
where A
l
, B
l

ν
= g
µν
. (3.1.48)
This means that any scalar product in the Minkowski space is invariant with respect to the
Lorentz transformations. Moreover, the scalar products of vectors (recall that the latter are
expressed through the
differences
in the coordinates of two points) are also invariant with
respect to the four-dimensional translations forming the Abelian (commutative) group T
4
.In
particular, the reader can easily verify that dx
µ
g
µν
dx
ν
and (∂/∂ x
µ
)g
µν
(∂/∂ x
ν
), where g
µν
denotes the inverse matrix
g
µρ
g

ϕ
2
(x)]. (3.1.49)
The action for a scalar relativistic field and for the entire time line −∞ < x
0
< ∞ can now be
written as follows:
S
0
[ϕ]=

4
d
4
x
0
( ˙ϕ,ϕ). (3.1.50)
Taking into account the fact that the integration measure d
4
x = dx
0
dx
1
dx
2
dx
3
is invariant with
respect to the pseudo-orthogonal Lorentz transformations as well as with respect to translations,
we can readily check that action (3.1.50) is indeed Poincar

2
dx
3
( ˙ϕ,ϕ). (3.1.51)
The equation of motion can now be derived from the extremality of action (3.1.51): δS = 0,
together with the boundary conditions that variations of the field at times t
0
and t
f
vanish:
δϕ(t
0
) = δϕ(t
f
) = 0, which result in the Euler–Lagrange equation
∂
∂t
δL
δ ˙ϕ
=
δL
δϕ
(3.1.52)
Path-integral formulation of the simplest quantum field theories
11
or
∂
∂t
∂
∂ ˙ϕ

. (3.1.55)
In order to describe
interacting
particles, we have to add, to the Lagrangian density (3.1.49),
higher powers of the field ϕ(x):
(∂
µ
ϕ,ϕ) =
1
2
[g
µν
∂
µ
ϕ(x)∂
µ
ϕ(x) − m
2
ψ
2
(x)]−V(ϕ(x)). (3.1.56)
Here the function V(ϕ(x)) describes a field self-interaction. The equation of motion for ϕ now
becomes
(
+ m
2
)ϕ(x) =−
∂V (ϕ)
∂ϕ
. (3.1.57)

satisfying
canonical anticommutation relations
(see any textbook on quantum field theory, e.g., Bogoliubov and Shirkov (1959), Bjorken and
Drell (1965) and Itzykson and Zuber (1980)).
A system of free relativistic spin-
1
2
fermions is described by a four-component complex field
ψ
α
(x), α = 1, ,4 and has the Lagrangian density
(x) =
¯
ψ(x)(i∂ − m)ψ(x) ≡
4

α,β=1
¯
ψ
α
(x)(i∂
αβ
− mδ
αβ
)ψ
β
(x) (3.1.59)
where we have introduced the standard notation: for any four-dimensional vector A
µ
,the

ν
γ
µ
= 2g
µν
1I
4
(3.1.61)
(1I
4
is the 4×4 unit matrix). In particular,∂ ≡ γ
µ
∂
µ
. One possible representation of the γ -matrices
has the form
γ
0
=

1I
2
0
0 −1I
2

γ
i
=


4

β=1
ψ
†
β
(x)γ
0
βα
. (3.1.63)
Note that, as is usual in the literature on quantum field theory, we do not use special print for
either the γ -matrices or the Pauli matrices (similarly to four-dimensional vectors).
The extremality condition for the action with the density (3.1.59) (Euler equation) gives the
Dirac equation
for a spin-
1
2
field
(i∂ − m)ψ(x) = 0. (3.1.64)
The general form for the expansion of a solution of the Dirac equation (3.1.64) over plane waves
is the following:
ψ(t, r) =
1
(2π)
3/2
2

i=1

d

†
i
(k)e
−ikx
+ c
∗
i
(k)v
†
i
(k)e
ikx
]
(3.1.65)
where k
0
= ω
k
≡
√
k
2
+ m
2
and u
i
(k), v
i
(k), i = 1, 2 comprise the complete set of orthonormal
solutions of the Dirac equation (in the momentum representation):

¯v
i
(k)v
j
(k) ≡
n

α=1
( ¯v
i
(k))
α
(v
j
(k))
α
=−¯u
i
(k)u
j
(k) =
m
ω
k
δ
ij
v
†
i
(k)v

(k))
β
− (u
i
(k))
α
( ¯u
i
(k))
β
]=
m
ω
k
δ
αβ

i
(v
i
(k))
α
( ¯v
i
(k))
β
=
1
2ω
k

, c
∗
i
into creation and annihilation
operators for fermionic particles. To take into account the Pauli principle, we have to impose
anticommutation relations
:
{

b
i
(k),

b
†
j
(k

)}=δ
ij
δ
3
(k − k

)
{c
i
(k),c
†
j

i
(k),

b
†
j
(k

)}={c
†
i
(k),c
†
j
(k

)}=0
{

b
i
(k),c
j
(k

)}={

b
i
(k),c

ψ
†
:
{
ˆ
ψ
α
(t, r),
ˆ
ψ
†
β
(t, r

)}=δ
αβ
δ
3
(r − r

)
{
ˆ
ψ
α
(t, r),
ˆ
ψ
β
(t, r

ψ
α
= iψ
†
α
. (3.1.70)
Thus the commutation relations (3.1.69) are nothing other than the fermionic generalization of
the canonical commutation relations for (generalized) coordinates and conjugate momenta. The
corresponding Hamiltonian
H =

d
3
r (π
˙
ψ − )
( is the Lagrangian density (3.1.59)) in the quantum case can be written in terms of the
fermionic creation and annihilation operators:

H =
2

i=1

d
3
k ω
k
[


k ω
k
[

b
†
i
(k)

b
i
(k) +c
†
i
(k)c
i
(k)].
Sometimes this procedure of energy subtraction in the fermionic case is carried out by
invoking the qualitative concept of a Dirac ‘sea’, i.e. we assume that all negative-energy states
are occupied and real experimentally observable particles correspond to excitations of this
background state of the whole fermion system (see e.g., Bjorken and Drell (1965)).
3.1.2 Path-integral representation for transition amplitudes in quantum field theories
In the operator approach, there exist different representations for the canonical field operators ˆϕ(r) and
ˆπ(r) satisfying the commutation relations (3.1.29) or (3.1.69). For example, in the case of the scalar field
theory and in the coordinate representation, the vectors of the corresponding Hilbert space of states are
functionals [ϕ(r)] of the field ϕ(t, r) and the operator ˆϕ is diagonal:
ˆϕ(r)[ϕ(r)]=ϕ(r)[ϕ(r)]
ˆπ(r)[ϕ(r)]= −i
δ
δϕ(r)

0
(t
0
, r), t
0
=ϕ(r)|e
−i(t−t
0
)

H
|ϕ
0
(r)
= lim
L→∞
lim
K→∞
a→0
lim
N→∞
ε→0
K

k=1

N

j=1


s
(r
l
), )} (3.1.72)
where the discrete-time and discrete-space approximated action S
N
depends on all π
i
(r
l
), i = 1, ,N +
1; l = 1, ,K and ϕ
s
(r
l
), s = 0, ,N + 1; l = 1, ,K variables (N is the number of time slices
and ε is the ‘distance’ between the time slices). In the case of Hamiltonian (3.1.28), the continuous limits
Path-integral formulation of the simplest quantum field theories
15
in (3.1.72) correspond to the following path integral:
ϕ(t, r), t|ϕ
0
(t
0
, r), t
0
=

{ϕ
0

i
ϕ(τ, r))
2
−
1
2
m
2
ϕ
2
(τ, r) − V (ϕ(τ, r))

. (3.1.73)
Gaussian integration over the momenta π(τ, r) yields the Feynman path integral in the coordinate space:
ϕ(t, r), t|ϕ
0
(t
0
, r), t
0
=
−1

{ϕ
0
(r),t
0
;ϕ(r),t}
dτ
ϕ(τ, r) exp


dxπ
2
(x)

(3.1.75)
is the normalization constant for expressing the transition amplitude via the Feynman (configuration) path
integral.
Expression (3.1.74) is almost invariant with respect to relativistic Poincar´e transformations. The
only source of non-invariance is the restriction of the time integral in the exponent to the finite interval
[t
0
, t]. However, it is necessary to point out that elementary particle experimentalists do not measure
probabilities directly related to amplitudes for transitions between eigenstates |ϕ(t
0
, r), t
0
 and |ϕ(t, r), t
of the quantum field ˆϕ(τ, r), but rather probabilities related to S-matrix elements, i.e. to probability
amplitudes for transitions between states which, at t →±∞, contain definite numbers of particles of
various types. These are called ‘in’ and ‘out’, |α, in and |β, out,whereα and β denote sets of quantum
numbers characterizing momenta, spin z-components and types of particle (e.g., photons, leptons, etc).
The S-matrix operator is defined as follows (cf (2.3.136)):

S = lim
t→+∞
t

→−∞
e

(see later).
♦ The path integral in holomorphic representation
The path-integral representation for transition amplitudes (3.1.74) is not particularly convenient for
deriving the matrix elements of the scattering operator (3.1.76). Even the path-integral expression for