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World Scientific Lecture Notes in Physics - Vol. 75
FIELD THEORY
A Path Integral Approa,
Second Edition
ASHOK DAS

FIELD THEORY
A Path Integral Approach
Second Edition

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World Scientific Lecture Notes in Physics - Vol. 75
FIELD THEORY
A Path Integral Approach
Second Edition
XV^ilUJx. JL/xk3
University of Rochester, USA
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Field theory a path integral approach 2nd Edition Ashok Das

Preface to the First Edition
Traditionally, field theory had its main thrust of development in high
energy physics. Consequently, the conventional field theory courses
are taught with a heavy emphasis on high energy physics. Over the
years, however, it has become quite clear that the methods and tech-
niques of field theory are widely applicable in many areas of physics.
The canonical quantization methods, which is how conventional field
theory courses are taught, do not bring out this feature of field the-
ory. A path integral description of field theory is the appropriate
setting for this. It is with this goal in mind, namely, to make gradu-
ate students aware of the applicability of the field theoretic methods
to various areas, that the Department of Physics and Astronomy at
the University of Rochester introduced a new one semester course on
field theory in Fall 1991.
This course was aimed at second year graduate students who had
already taken a one year course on nonrelativistic quantum mechan-
ics but had not necessarily specialized into any area of physics and
these lecture notes grew out of this course which I taught. In fact,
the lecture notes are identical to what was covered in the class. Even
in the published form, I have endeavored to keep as much of the de-
tailed derivations of various results as I could — the idea being that
a reader can then concentrate on the logical development of concepts
without worrying about the technical details. Most of the concepts
were developed within the context of quantum mechanics — which
the students were expected to be familiar with — and subsequently
these concepts were applied to various branches of physics. In writ-
ing these lecture notes, I have added some references at the end of
vii

viii Field Theory: A Path Integral Approach
every chapter. They are only intended to be suggestive. There is so
much literature that is available in this subject that it would have
been impossible to include all of them. The references are not meant
to be complete and I apologize to many whose works I have not cited
in the references. Since this was developed as a course for general
students, the many interesting topics of gauge theories are also not
covered in these lectures. It simply would have been impossible to
do justice to these topics within a one semester course.
There are many who were responsible for these lecture notes. I
would like to thank our chairman, Paul Slattery, for asking me to
teach and design a syllabus for this course. The students deserve
the most credit for keeping all the derivations complete and raising
many issues which I, otherwise, would have taken for granted. I am
grateful to my students Paulo Bedaque and Wen-Jui Huang as well
as to Dr. Zhu Yang for straightening out many little details which
were essential in presenting the material in a coherent and consistent
way. I would also like to thank Michael Begel for helping out in
numerous ways, in particular, in computer-generating all the figures
in the book. The support of many colleagues was also vital for the
completion of these lecture notes. Judy Mack, as always, has done
a superb job as far as the appearance of the book is concerned and
I sincerely thank her. Finally, I am grateful to Ammani for being
there.
Ashok Das,
Rochester.

Preface to the Second Edition
This second edition of the book is an expanded version which con-
tains a chapter on path integral quantization of gauge theories as well
as a chapter on anomalies. In addition, chapter 6 (Supersymmetry)
has been expanded to include a section on supersymmetric singular
potentials. While these topics were not covered in the original course
on path integrals, they are part of my lectures in other courses that
I have taught at the University of Rochester and have been incorpo-
rated into this new edition at the request of colleagues from all over
the world. There are many people who have helped me to complete
this edition of the book and I would like to thank, in particular, Judy
Mack, Arsen Melikyan, Dave Munson and J. Boersma for all their
assistance.
Ashok Das,
Rochester.

Contents
Preface to the First Edition vii
Preface to the Second Edition ix
1. Introduction 1
1.1 Particles and Fields 1
1.2 Metric and Other Notations 2
1.3 Functionals 3
1.4 Review of Quantum Mechanics 7
1.5 References 10
2. Path Integrals and Quantum Mechanics 11
2.1 Basis States 11
2.2 Operator Ordering 13
2.3 The Classical Limit 20
2.4 Equivalence with the Schrodinger Equation 22
2.5 Free Particle 25
2.6 References 30
3. Harmonic Oscillator 31
3.1 Path Integral for the Harmonic Oscillator 31
3.2 Method of Fourier Transform 33
3.3 Matrix Method 36
3.4 The Classical Action 45
xi

xii Field Theory: A Path Integral Approach
3.5 References 51
4. Generating Functional 53
4.1 Euclidean Rotation 53
4.2 Time Ordered Correlation Functions 59
4.3 Correlation Functions in Definite States 61
4.4 Vacuum Functional 64
4.5 Anharmonic Oscillator 71
4.6 References 73
5. Path Integrals for Fermions 75
5.1 Fermionic Oscillator 75
5.2 Grassmann Variables 78
5.3 Generating Functional 83
5.4 Feynman Propagator 86
5.5 The Fermion Determinant 91
5.6 References 95
6. Supersymmetry 97
6.1 Supersymmetric Oscillator 97
6.2 Supersymmetric Quantum Mechanics 102
6.3 Shape Invariance 105
6.4 Example 110
6.5 Supersymmetry and Singular Potentials Ill
6.5.1 Regularized Superpotential 115
6.5.2 Alternate Regularization 117
6.6 References 118
7. Semi-Classical Methods 121
7.1 WKB Approximation 121
7.2 Saddle Point Method 127
7.3 Semi-Classical Methods in Path Integrals 130
7.4 Double Well Potential 134
7.5 References 142

Contents xiii
8. Path Integral for the Double Well 143
8.1 Instantons 143
8.2 Zero Modes 150
8.3 The Instanton Integral 154
8.4 Evaluating the Determinant 158
8.5 Multi-Instanton Contributions 163
8.6 References 166
9. Path Integral for Relativistic Theories 167
9.1 Systems with Many Degrees of Freedom 167
9.2 Relativistic Scalar Field Theory 170
9.3 Feynman Rules 181
9.4 Connected Diagrams 184
9.5 References 186
10. Effective Action 187
10.1 The Classical Field 187
10.2 Effective Action 193
10.3 Loop Expansion 200
10.4 Effective Potential at One Loop 203
10.5 References 208
11. Invariances and Their Consequences 209
11.1 Symmetries of the Action 209
11.2 Noether's Theorem 212
11.2.1 Example 215
11.3 Complex Scalar Field 218
11.4 Ward Identities 222
11.5 Spontaneous Symmetry Breaking 226
11.6 Goldstone Theorem 235
11.7 References 236
12. Gauge Theories 239
12.1 Maxwell Theory 239
12.2 Non-Abelian Gauge Theory 246
12.3 Path Integral for Gauge Theories 255

xiv Field Theory: A Path Integral Approach
12.4 BRST Invariance 266
12.5 Ward Identities 274
12.6 References 278
13. Anomalies 279
13.1 Anomalous Ward Identity 279
13.2 Schwinger Model 289
13.3 References 307
14. Systems at Finite Temperature 309
14.1 Statistical Mechanics 309
14.2 Critical Exponents 314
14.3 Harmonic Oscillator 318
14.4 Fermionic Oscillator 324
14.5 References 326
15. Ising Model 327
15.1 One Dimensional Ising Model 327
15.2 The Partition Function 332
15.3 Two Dimensional Ising Model 337
15.4 Duality 339
15.5 High and Low Temperature Expansions 343
15.6 Quantum Mechanical Model 349
15.7 Duality in the Quantum System 356
15.8 References 358
Index 359

Chapter 1
Introduction
1.1 Particles and Fields
Classically, there are two kinds of dynamical systems that we en-
counter. First, there is the motion of a particle or a rigid body (with
a finite number of degrees of freedom) which can be described by a
finite number of coordinates. And then, there are physical systems
where the number of degrees of freedom is nondenumerably (non-
countably) infinite. Such systems are described by fields. Familiar
examples of classical fields are the electromagnetic fields described by
E(x, t) and B(x, t) or equivalently by the potentials (0(x, t), A(x, t)).
Similarly, the motion of a one-dimensional string is also described by
a field 4>(x,t), namely, the displacement field. Thus, while the coor-
dinates of a particle depend only on time, fields depend continuously
on some space variables as well. Therefore, a theory described by
fields is usually known as a D + 1 dimensional field theory where D
represents the number of spatial dimensions on which the field vari-
ables depend. For example, a theory describing the displacements
of the one-dimensional string would constitute a 1+1 dimensional
field theory whereas the more familiar Maxwell's equations (in four
dimensions) can be regarded as a 3+1 dimensional field theory. In
this language, then, it is clear that a theory describing the motion of
a particle can be regarded as a special case, namely, we can think of
such a theory as a 0+1 dimensional field theory.
1

2 Field Theory: A Path Integral Approach
1.2 Metric and Other Notations
In these lectures, we will discuss both non-relativistic as well as rel-
ativists theories. For the relativistic case, we will use the Bjorken-
Drell convention. Namely, the contravariant coordinates are assumed
to be
x» = (t,x), /i = 0,l,2,3, (1.1)
while the covariant coordinates have the form
X/j, = T],MVX'/
= (t, - x ) . (1.2)
Here we have assumed the speed of light to be unity (c = 1). The
covariant metric, therefore, follows to have a diagonal form with the
signatures
Vnu = ( + , - , - , - ) • (1.3)
The inverse or the contravariant metric clearly also has the same
form, namely,
TT = ( + , - , - , - ) • (1.4)
The invariant length is given by
x2
= x% = rTxpXv = ri^x" = t2
- x2
. (1.5)
The gradients are similarly obtained from Eqs. (1.1) and (1.2) to be
a
-=aMf'v
)' (L6)
<"=£=(*•-*)• <17)
so that the D'Alembertian takes the form
• = d^ = rrd»dv = ~-V2
. (1.8)

Introduction 3
1.3 Functionals
It is evident that in dealing with dynamical systems, we are dealing
with functions of continuous variables. In fact, most of the times,
we are really dealing with functions of functions which are otherwise
known as functionals. If we are considering the motion of a particle
in a potential in one dimension, then the Lagrangian is given by
L(x, x) = - m i 2
- V(x), (1.9)
where x(t) and x(t) denote the coordinate and the velocity of the
particle and the simplest functional we can think of is the action
functional defined as
S[x] = [ f
dtL(x,x). (1.10)
Note that unlike a function whose value depends on a particular
point in the coordinate space, the value of the action depends on
the entire trajectory along which the integration is carried out. For
different paths connecting the initial and the final points, the value
of the action functional will be different.
Thus, a functional has the generic form
Ff] = JdxF(f(x)), (1.11)
where, for example, we may have
F(f(x)) = (f(x))n
. (1.12)
Sometimes, one loosely also says that F(f(x)) is a functional. The
notion of a derivative can be extended to the case of functionals in a
natural way through the notion of generalized functions. Thus, one
defines the functional derivative or the Gateaux derivative from the
linear functional
F'v]=^-F{f + ev] =
/ d l
^ " W
- ( U 3 )

Equivalently, from the working point of view, this simply corresponds
to defining
Sf(y) e - o €
l
' '
It now follows from Eq. (1.14) that
Sf{x)
Sf(y)
8(x-y). (1.15)
The functional derivative satisfies all the properties of a deriva-
tive, namely, it is linear and associative,
5
•(FM+FM=m+mm
5f(XyllJi ZUiJ
5f{x) 6f(x)'
5
<F1[f]F2[f]) = 5
-^4F2[f} + F1[f}S
^4. (1.16)
6f{xy xWJ ,uu Sf{x) . W J x W J 5f{x)
It also satisfies the chain rule of differentiation. Furthermore, we now
see that given a functional F[f], we can Taylor expand it in the form
F[f] = fdx P0(x) + fdXldx2 Pi(xux2) f(x2)
+ dx1dx2dx3P2(xi,x2,x3) f(x2)f(x3)- , (1.17)
where
Pi(xi,x2) =
P2(xi,x2,x3) =
P0{x) = F(f(x))nx)=0 ,
8F(f(xi))
Sf(x2)
1 PFifixt))
(1.18)
f(x)=0
2! 5f(x2)8f(x3) /(x)=0
and so on.
As simple examples, let us calculate a few particular functional
derivatives.

Introduction 5
(i) Let
F[f] = Jdy F(f(y)) = J dy (f(y))n
, (1.19)
where n denotes a positive integer. Then,
SF(f(y)) = H m F(f(y) + e5(y-x))-F(f(y))
Sf(x) e™ e
l]m(f(y) + e5(y-x))n
-(f(y))n
= H m ( / ( y ) ) n + n
<^y))n
~l
^y - *) + °(e2
) - (f(y»n
= n ( / ( y ) ) n _ 1
% - a ; ) . (1.20)
Therefore, we obtain
SFf] _ f 5F(f(y))
6f(x) J V
8f(x)
= Jdyn(f(y)r-1
5(y-x)
= n(f(x))n
~1
. (1.21)
(ii) Let us next consider the one-dimensional action in Eq. (1.10)
S[x] = [ f
dt' L(x(t'),x(t')), (1.22)
with
L(x(t),x(t)) = m{x{t)f -V(x(t))
= T{x(t))-V(x(t)). (1.23)
In a straightforward manner, we obtain
SV(x(t')) = y(s(f) + e£ft' - t)) - V{x(t'))
Sx(t) e^o e
= ^'(x^'))*^ - *), (1-24)

where we have defined
Similarly,
ST(x(t')) = U m T(x(t>) + e£r5(t>-t))-T(x(t>))
5x(t) e^O €
= mx(t')—5(t'-t). (1.25)
It is clear now that
SL(x(t'),x(t')) _ 5{T{x(t'))-V{x{t')))
8x(t) ~ Sx(t)
= mx(t') -^6(1/ -t)- V'{x(t'))5{t' - i). (1.26)
Consequently, in this case, we obtain for ti <t <tf
5S[x] = /•*/ 5L(x(t'),x(t'))
8x(t) Jti 6x(t)
= ff
dt' {mx{t')^-/8(t' -t)- V'(x{t'))5{t' - t))
Jti dt
= -mx(t) - V'(x(t))
ddL(x(t),x(t)) dL(x(t),x(t))
dt dx{t) +
dx(t) ' {
' '
The right hand side is, of course, reminiscent of the Euler-Lagrange
equation. In fact, we note that
6x(t) dt 8x(t) 8x{t) ' K
' '
gives the Euler-Lagrange equation as a functional extremum of the
action. This is nothing other than the principle of least action ex-
pressed in a compact notation in the language of functionals.

Introduction 7
1.4 Review of Quantum Mechanics
In this section, we will describe very briefly the essential features of
quantum mechanics assuming that the readers are familiar with the
subject. The conventional approach to quantum mechanics starts
with the Hamiltonian formulation of classical mechanics and pro-
motes observables to non-commuting operators. The dynamics, in
this case, is given by the time-dependent Schrodinger equation
* * ! * £ » _ H h W > , ( 1 3 )
where H denotes the Hamiltonian operator of the system. Equiva-
lently, in the one dimensional case, the wave function of a particle
satisfies
h2
d2

+ V(x))^(x,t), (1.30)
2m dx2
where we have identified
# E , * ) = <*#(*)>. (1-31)
with x) denoting the coordinate basis states. This, then, defines the
time evolution of the system.
The main purpose behind solving the Schrodinger equation lies
in determining the time evolution operator which generates the time
translation of the system. Namely, the time evolution operator trans-
forms the quantum mechanical state at an earlier time ti to a future
time t as
M*l)) = tf(*l,*2M*2)>. (1-32)
Clearly, for a time independent Hamiltonian, we see from Eq. (1.29)
(the Schrodinger equation) that for t > t2,
U(h,t2) = e-^tl
-t2)H
. (1.33)

More explicitly, we can write
U(h,t2) = B(h - t2)e-^-^H
. (1.34)
It is obvious that the time evolution operator is nothing other than
the Green's function for the time dependent Schrodinger equation
and satisfies
(ih-^- - H U(h,t2) = ih5(h - *2) • (1-35)
Determining this operator is equivalent to finding its matrix elements
in a given basis. Thus, for example, in the coordinate basis defined
by
Xx)=xx), (1.36)
we can write
{xiU(t1,t2)x2) = U(ti,x1;t2,x2). (1-37)
If we know the function U(ti,xit2,x2) completely, then the time
evolution of the wave function can be written as
ip(xi,h) = dx2U{ti,xi;t2,x2)ip(x2,t2). (1.38)
It is interesting to note that the dependence on the intermediate
times drops out in the above equation as can be easily checked.
Our discussion has been within the framework of the Schr-odinger
picture so far where the quantum states i/j(t)) carry time dependence
while the operators are time independent. On the other hand, in the
Heisenberg picture, where the quantum states are time independent,
using Eq. (1.32) we can identify
IV>H = m = o))s = m = o))
(1.39)

Introduction 9
In this picture, the operators carry all the time dependence. For
example, the coordinate operator in the Heisenberg picture is related
to the coordinate operator in the Schrodinger picture through the
relation
XH{t) = e*tH
Xe-*tH
. (1.40)
The eigenstates of this operator satisfying
XH{t)x,t)H = xx,t)H, (1.41)
are then easily seen to be related to the coordinate basis in the
Schrodinger picture through
x,t)H = e*tH
x). (1.42)
It is clear now that for t > t2 we can write
H(xi,t1x2,t2)H = ( m l e - i ^ e t * 2
^ )
= (xle-^-t
^H
x2)
= {xiU{h,t2)x2)
= U(t1,x1;t2,x2). (1-43)
This shows that the matrix elements of the time evolution opera-
tor are nothing other than the time ordered transition amplitudes
between the coordinate basis states in the Heisenberg picture.
Finally, there is the interaction picture where both the quantum
states as well as the operators carry partial time dependence. With-
out going into any technical detail, let us simply note here that the
interaction picture is quite useful in the study of nontrivially inter-
acting theories. In any case, the goal of the study of quantum me-
chanics in any of these pictures is to construct the matrix elements of
the time evolution operator which as we have seen can be identified
with transition amplitudes between the coordinate basis states in the
Heisenberg picture.

1.5 References
Das, A., "Lectures on Quantum Mechanics", Hindustan Book
Agency.
Dirac, P. A. M., "Principles of Quantum Mechanics", Oxford Univ.
Press.
Schiff, L. I., "Quantum Mechanics", McGraw-Hill Publishing.

Chapter 2
Path Integrals and Quantum
Mechanics
2.1 Basis States
Before going into the derivation of the path integral representation
for U(tf,xf,ti,Xi) or the transition amplitude, let us recapitulate
some of the basic formulae of quantum mechanics. Consider, for sim-
plicity, a one dimensional quantum mechanical system. The eigen-
states of the coordinate operator, as we have seen in Eq. (1.36),
satisfy
Xx)=xx). (2.1)
These eigenstates define an orthonormal basis. Namely, they satisfy
(xx') = 5(x — x'),
dxx)(x = l. (2.2)
Similarly, the eigenstates of the momentum operator satisfying
Pp)=pp), (2.3)
also define an orthonormal basis. Namely, the momentum eigenstates
satisfy
(pp') = 8(p-p>),
Jdpp){p = l. (2.4)
l l

The inner product of the coordinate and the momentum basis states
gives the matrix elements of the transformation operator between
the two basis. In fact, one can readily determine that
(px) = ^ e " ^ = (xW . (2.5)
These are the defining relations for Fourier transforms. Namely, us-
ing the completeness relations of the basis states, the Fourier trans-
form of functions can be defined as
f(x) = {xf) = Jdp{xp){pf)
1 [
V^ J '
dk el
"x
f{k), (2.6)
f(k) = Vhf(p)
f dxe~y;px
f(x)
-}=Jdxe-^f(x). (2.7)
These simply take a function from a given space to its conjugate
space or the dual space. Here k = | can be thought of as the wave
number in the case of a quantum mechanical particle. (Some other
authors may define Fourier transform with alternate normalizations.
Here, the definition is symmetrical.)
As we have seen in Eq. (1.42), the Heisenberg states are related
to the Schrodinger states in a simple way. For the coordinate basis
states, for example, we will have
x,t)H = e%m
x).
It follows now that the coordinate basis states in the Heisenberg

Path Integrals and Quantum Mechanics 13
picture satisfy
^tHAtHJ
H(x,tx',t)H = (xe-Kth
e*tH
x')
= (xx') = 5(x - x1
), (2.8)
and
' tH
dx x,t}H H(x,t = dxehtH
x)(
= e-nm
fdxx){
xe h
, xe~*tH
= 1. (2.9)
It is worth noting here that the orthonormality as well as the com-
pleteness relations hold for the Heisenberg states only at equal times.
2.2 Operator Ordering
In the Hamiltonian formalism, the transition from classical mechan-
ics to quantum mechanics is achieved by promoting observables to
operators which are not necessarily commuting. Consequently, the
Hamiltonian of the classical system is supposed to go over to the
quantum operator
H(x,p)^H(xop,Pop). (2.10)
This, however, does not specify what should be done when products
of x and p (which are non-commuting as operators) are involved. For
example, classically we know that
xp — px.
Therefore, the order of these terms does not matter in the classical
Hamiltonian. Quantum mechanically, however, the order of the op-
erators is quite crucial and a priori it is not clear what such a term

ought to correspond to in the quantum theory. This is the oper-
ator ordering problem and, unfortunately, there is no well defined
principle which specifies the order of operators in the passage from
classical to quantum mechanics. There are, however, a few prescrip-
tions which one uses conventionally. In normal ordering, one orders
the products of x's and p's such that the momenta stand to the left
of the coordinates. Thus,
N.O.
xp —> px ,
N.O.
px —• px ,
2 N.O. 2
x p —• px ,
N.O. o /r. 1 1
xpx —> px , (2.11)
and so on. However, the prescription that is much more widely used
and is much more satisfactory from various other points of view is
the Weyl ordering. Here one symmetrizes the product of operators
in all possible combinations with equal weight. Thus,
xp —^ ~{xp + px),
w.o. I , , .
px —• -(xp + px),
2 W.O. I / 9 , , 2
x p —> -[x p + xpx + px ) ,
xpx —>' -(x2
p + xpx + px2
), (2-12)
and so on.
For normal ordering, it is easy to see that for any quantum Hamil-
tonian obtained from the classical Hamiltonian H(x,p)
(x'HN
-°-x) = f dp(x'p)(pHN
-°-x)
= J' ^.e-&*-^H{x,p). (2.13)

Here we have used the completeness relations of the momentum basis
states given in Eq. (2.4) as well as the defining relations in Eqs. (2.1),
(2.3) and (2.5). (The matrix element of the quantum Hamiltonian is a
classical function for which the ordering is irrelevant.) To understand
Weyl ordering, on the other hand, let us note that the expansion of
(axop + j3pop)N
,
generates the Weyl ordering of products of the form a^opPop naturally
if we treat xop and pop as non-commuting operators. In fact, we can
easily show that
(*xop + (3pop)N
= ] T - ^ an
r(xn
opP™)W
-°- (2.14)
n+m=N
The expansion of the exponential operator
e(aa;op+/3pop)
would, of course, generate all such powers and by analyzing the ma-
trix elements of this exponential operator, we will learn about the
matrix elements of Weyl ordered Hamiltonians.
Prom the fact that the commutator of a;op and pop is a constant,
we obtain using the Baker-Campbell-Hausdorff formula
Q 2 )ePPope( 2 I = gl 2 ) (APPop~ 2 ~ >
— e(ax0p+/3p0p) _ (2.15)
Using this relation, it can now be easily shown that
dp (x e(
2 >epp
°p
p){peK
2 >x)
/
p-e-fc*-* C-^+0v). (2.16)

Once again, we have used here the completeness properties given in
Eq. (2.4) as well as the defining relations in Eqs. (2.1), (2.3) and (2.5).
It follows from this that for a Weyl ordered quantum Hamiltonian,
we will have
< z ' | # w
- ° - ( W o p ) | , ) = f ^ e-i«°-*H {^f,p) • (2.17)
As we see, the matrix elements of the Weyl ordered Hamiltonian leads
to what is known as the mid-point prescription and this is what we
will use in all of our discussions.
We are now ready to calculate the transition amplitude. Let us
recall that in the Heisenberg picture, for tf > U, we have
U (tf,Xf]ti,Xi) = H(Xf,tfXi,ti)H.
Let us divide the time interval between the initial and the final time
into N equal segments of infinitesimal length e. Namely, let
* = tj
jr • (2.i8)
In other words, for simplicity, we discretize the time interval and
in the end, we are interested in taking the continuum limit e —
> 0
and N —
> oo such that Eq. (2.18) holds true. We can now label the
intermediate times as, say,
tn = U + ne, n = l , 2 , . . . , ( i V - l ) . (2.19)
Introducing complete sets of coordinate basis states for every in-
termediate time point (see Eq. (2.9)), we obtain
U(tf,xf,ti,Xi) = H(xf,tfxi,ti)H
= lim dx1---dxN-lH{xf,tfxN-i,tN-i)H
N-^oo
X H{XN-l-,tN-lxN-2,tN-2)H • • -H {xi,hXi,ti)H .
(2.20)

In writing this, we have clearly assumed an inherent time ordering
from left to right. Let us also note here that while there are iV inner
products in the above expression, there are only (N— 1) intermediate
points of integration. Furthermore, we note that any intermediate
inner product in Eq. (2.20) has the form
H%nitnxn-i,tn—i)H = xne ft eft xn—)
= { a g e - ^ - ' - ^ l x n - i )
= (xne~^eH
xn-i)
= f d
Pn JpnjXn-Xn^-^H^-^^^) ^ ^ ^
Here we have used the mid-point prescription of Eq. (2.17) corre-
sponding to Weyl ordering.
Substituting this form of the inner product into the transition
amplitude, we obtain
U(tf,xf,ti,Xi)= lim / dxi • • • dajjv-i
e->0 J
dpi dpN
2-Kh'" 2nh
N—>oo
x e i E ^ K . » - 0 - « f i ( = ^ * . ) ) _ (2.22)
In writing this, we have identified
XQ = Xi, XN = Xf. (2.23)
This is the crudest form of Feynman's path integral and is defined
in the phase space of the system. It is worth emphasizing here that
the number of intermediate coordinate integrations differs from the
number of momentum integrations and has profound consequences
in the study of the symmetry properties of the transition amplitudes.
Note that in the continuum limit, namely, for e —
> 0, we can write

the phase factor of Eq. (2.22) as
N
I™ ^ Yl [P^(x
n ~x
n-i) -eH ( ^n
' 1
, p n ) )
N^oo " n=1
N
/ /
i. ^ ^ v
/ / Xn Xn— 1 TT I %n ~r Xn—
hm - e £ [Pn [ J - H ( ,Pr
=
-*£dtip±
=-[
H(x,p))
dt L. (2.24)
U
Namely, it is proportional to the action in the mixed variables.
To obtain the more familiar form of the path integral involving the
Lagrangian in the configuration space, let us specialize to the class
of Hamiltonians which are quadratic in the momentum variables.
Namely, let us choose
H(x,p) = ^ + V(x). (2.25)
n such a case, we have from Eq. (2.22)
U(tf,xf,ti,Xi) = hm / dzi • • • dxN-i„ fc •
N—>oo
x e V v
dpN
2-KK
y/xn+xn.
*)). (2.26)
The momentum integrals are Gaussian and, therefore, can be done
readily. We note that
/
A„ icfPn Pn<.x„-xn_1)
U
Pn t I 2m £
2TT^
6
/
An i£ ( „1 2m
Pn(xn-xn-l)
a
Pn
e-2^h{Pn i
2irh

" / 2nE e
m(xn-xn_i)ji
( m(xn-xn_i) '
'-)-(•-
1 (2'Kmh2
gif1
""1
"-1
= ( — Y
2mheJ
(2.27)
Substituting this back into the transition amplitude in Eq. (2.26),
we obtain
N_
U(tf,xf;tl,xi) = lim ( ^ ) 2
/
iey^AT I m fx
n-x
n-l y(Xn+x
n-l
dxi • • • dxN-ieh n _ 1
V 2
^ e
' ^
= A f Vx etti<l
*m±2
-V
^
= AJvxe*s
W, (2.28)
where A is a constant independent of the dynamics of the system
and S[x] is the action for the system given in Eq. (1.10). This is
Feynman's path integral for the transition amplitude in quantum
mechanics.

To understand the meaning of this, let us try to understand the
meaning of the path integral measure Vx. In this integration, the
end points are held fixed and only the intermediate coordinates are
integrated over the entire space. Any spatial configuration of the
intermediate points, of course, gives rise to a trajectory between the
initial and the final points. Thus, integrating over all such configu-
rations (that is precisely what the integrations over the intermediate
points are supposed to do) is equivalent to summing over all the paths
connecting the initial and the final points. Therefore, Feynman's
path integral simply says that the transition amplitude between an
initial and a final state is the sum over all paths, connecting the
two points, of the weight factor eft ^J. We know from the study
of quantum mechanics that if a process can take place in several
distinct ways, then the transition amplitude is the sum of the indi-
vidual amplitudes corresponding to every possible way the process
can take place. The sum over the paths is, therefore, quite expected.
However, it is the weight factor e^s
^x
' that is quite crucial and un-
expected. Classically, we know that it is the classical action that
determines the classical dynamics. Quantum mechanically, however,
what we see is that all the paths contribute to the transition ampli-
tude. It is also worth pointing out here that even though we derived
the path integral representation for the transition amplitude for a
special class of Hamiltonians, the expression holds in general. For
Hamiltonians which are not quadratic in the momenta, one should
simply be careful in defining the path integral measure Vx.
2.3 The Classical Limit
As we have seen in Eq. (2.28), the transition amplitude can be written
as a sum over paths and for the case of a one dimensional Hamiltonian
which is quadratic in the momentum, it is represented as
U(tf,xf;ti,xi) = A f Vxei5
^ (2.29)
mfxn-xn-l2
y(x
n+x
n-l
e^O " J
JV—xx>
/
lim A , rl'ri . . . H T A r 1 P

where
AN
= (^k)2
•
Even though one can be more quantitative in the discussion of the
behavior of the transition amplitude, let us try to be qualitative in
the following. We note that for paths where
Xn ^ Xn—X,
the first term in the exponential would be quite large, particularly
since e is infinitesimally small. Therefore, such paths will lead to a
very large phase and consequently, the weight factor can easily be
positive or negative. In other words, for every such xn, there would
be a nearby xn differing only slightly which would have a cancelling
effect. Thus, in the path integral, all such contributions will average
out to zero.
Let us, therefore, concentrate only on paths connecting the initial
and the final points that differ from one another only slightly. For
simplicity, we only look at continuous paths which are differentiable.
(A more careful analysis shows that the paths which contribute non-
trivially are the continuous paths which are not necessarily differen-
tiable. But for simplicity of argument, we will ignore this technical
point.) The question that we would like to understand is how among
all the paths which can contribute to the transition amplitude, it
is only the classical path that is singled out in the classical limit,
namely, when h —
> 0. We note here that the weight factor in the

path integral, namely, eft *-x
is a phase multiplied by a large quan-
tity when h —
• 0. Mathematically, therefore, it is clear that the
dominant contribution to the path integral would arise from paths
near the one which extremizes the phase factor. In other words, only
the trajectories close to the ones satisfying
iM=0
' <
2
-3
0
)
would contribute significantly to the transition amplitude in the clas-
sical limit. But, from the principle of least action, we know that these
are precisely the trajectories which a classical particle would follow,
namely, the classical trajectories. Once again, we can see this more
intuitively in the following way. Suppose, we are considering a path,
say #3, which is quite far away from the classical trajectory. Then,
because h is small, the phase along this trajectory will be quite large.
For every such path, there will be a nearby path, infinitesimally close,
say #2, where the action would differ by a small amount, but since
it is multiplied by a large constant would produce a large phase. All
such paths, clearly, will average out to zero in the sum. Near the clas-
sical trajectory, however, the action is stationary. Consequently, if
we choose a path infinitesimally close to the classical path, the action
will not change. Therefore, all such paths will add up coherently and
give the dominant contribution as h —
» 0. It is in this way that the
classical trajectory is singled out in the classical limit, not because
it contributes the most, but rather because there are paths infinites-
imally close to it which add coherently. One can, of course, make
various estimates as to how far away a path can be from the classical
trajectory before its contribution becomes unimportant. But let us
not go into these details here.
2.4 Equivalence with the Schrodinger Equation
At this point one may wonder about the Schrodinger equation in the
path integral formalism. Namely, it is not clear how we can recover
the time dependent Schrodinger equation (see Eq. (1.30)) from the

path integral representation of the transition amplitude. Let us recall
that the Schrodinger equation is a differential equation. Therefore, it
determines infinitesimal changes in the wave function. Consequently,
to derive the Schrodinger equation, we merely have to examine the
infinitesimal form of the transition amplitude or the path integral.
Prom the explicit form of the transition amplitude in Eq. (2.29), we
obtain for infinitesimal e
U(tf = e,xf,ti = 0,Xi)
/m^Kl^'-vpP)). (M1)
2-Kihe) v
'
We also know from Eq. (1.38) that the transition amplitude is the
propagator which gives the propagation of the wave function in the
following way,
/
oo
dx' U(e,x;0,x')il>(x',0). (2.32)
-oo
Therefore, substituting the form of the transition amplitude namely,
Eq. (2.31) into Eq. (2.32), we obtain
1>(x, e) = ( ^ | - ) * | ° ° Me&l*-*'^f V
^H{x 0). (2.33)
Let us next change variables to
r] = x' - x , (2.34)
so that we can write
rP(x, e) = ( ~ ) h
J°° dV e S ^ - ^ ^ + l ) ] ^ , +v, 0). (2.35)
It is obvious that because e is infinitesimal, if 7
7 is large, then the first
term in the exponent would lead to rapid oscillations and all such
contributions will average out to zero. The dominant contribution
will, therefore, come from the region of integration
0 < H < ( ^ ) (2.36)

where the change in the first exponent is of the order of unity. Thus,
we can Taylor expand the integrand and since we are interested in
the infinitesimal behavior, we can keep terms consistently up to order
e. Therefore, we obtain
x U(x, 0) + rtfix, 0) + tl/'{x, 0) + 0{rf)
= (^)T*i
im 2
a Ihe 'I
-oo
il>{x,0) --V(x)1>(x,0)
,2
(2.37)
+ # ' ( * , 0) + |-<(z,0)+O(r73
,e
2
)
The individual integrations can be easily done and the results are
f°° , iEL„2 f2mhe*
/ dr? e^ne7
! =
J-oo V ™ /
/
oo
drirje^2
= 0, (2.38)
•oo
/•°° , 9 inL„2 ihe f2mhe2
I dr? 7
7 ezfie'7
= —
J-oo m m J
Note that these integrals contain oscillatory integrands and the
simplest way of evaluating them is through a regularization. For
example,
/
OO /"OO
dr? e ^ 7
' = lim / dry e^zsi"
.oo <5^0+ J_00
(, V = (™**) (2.39)
-OO
1 . . 1
= lim
<5->0+
and so on.

Substituting these back into Eq. (2.37), we obtain
- ( — J ^"(x,0) + O(e
2
)
=tf>(x, 0) + ^ V (x, 0) - jV (x) i, (x, 0) + O (e2
)
o r ^ ( x , £ ) - ^ ( x , 0 ) = - | ( - ^ ^ + y ( a ; ) ) v ( x , 0 ) + O(62
).
(2.40)
In the limit e —
• 0, therefore, we obtain the time dependent
Schrodinger equation (Eq. (1.30))
The path integral representation, therefore, contains the Schrodinger
equation and is equivalent to it.
2.5 Free Particle
We recognize that the path integral is a functional integral. Namely,
the integrand which is the phase factor is a functional of the
trajectory between the initial and the final points. Since we do not
have a feeling for such quantities, let us evaluate some of these inte-
grals associated with simple systems. The free particle is probably
the simplest of quantum mechanical systems. For a free particle in
one dimension, the Lagrangian has the form
L = - m i 2
. (2.41)

Therefore, from our definition of the transition amplitude in
Eq. (2.28) or (2.29), we obtain
U(tf,xf,ti,Xi)
= lim I ——-) / dxi • • • d ^ - i e h
^n = 1 2
^ £
)
e^o V 2mne I J
JV—>oo
lim ( ) 2
/ dxi • • •dxN-ie^'^n
=l(
-Xn
~Xn
e->o v 2-nine J J
i ) 2
TV—>oo
(2.42)
Defining
/ m
2heJ
we have
^ = U i d * « , (2-43)
U(tf,xf;U,Xi)= hm ( _ ) ^ _ J
—>oo
/"dyi • • -dj/jv-i e*E^=i(i/n-vn-i)a
. (2.44)
iV->oo
X
This is a Gaussian integral which can be evaluated in many different
ways. However, the simplest method probably is to work out a few
lower order ones and derive a pattern. We note that
[dyx e*[(w-i»)2
+(i«-yi)a
] = [dyi eiPfoi-^T^+^w-w)2
]
™*etto-vo) ( 2 .4 5 )

If we had two intermediate integrations, then we will have
/ dyidy2 ei
[^1
~yo
^2+
^2
~?/1
^+
^3
'"?/2
^2
]
= (*I* fdy2eih(y2-y0)2
+(y3-y2?]
) ' /
d y 2 e ifte-^)2
+t(«)2
2 / V 3 '
(wr
i
, 2 2
ef(»/3-!/o) (2.46)
A pattern is now obvious and using this we can write
U(tf,xf,ti,Xi)
e-+o 2irihe) m N
e
JV—>oo
h
m f^-)f
f^V
_l
e-^o 2mne/ m I fl
e
N—*oo
e2HNc (xN—xo)
= h m ( ) e^hNeix
f x
i)
e^o 2mhNeJ
AT—>oo
m
2irih(tf — ti)
I i m
(x
f-x
i)
a
(*/-*0 (2.47)
Thus, we see that for a free particle, the transition amplitude can be
explicitly evaluated. It has the right behavior in the sense that, we
see as tf —> U,
U{tf,xf,ti,Xi)->8(xf-Xi), (2.48)

which is nothing other than the orthonormality relation for the
states in the Heisenberg picture given in Eq. (2.8). Second, all the
potentially dangerous singular terms involving e have disappeared.
Furthermore this is exactly what one would obtain by solving the
Schrodinger equation. It expresses the well known fact that even
a well localized wave packet spreads with time. That is, even the
simplest of equations has only dispersive solutions.
Let us note here that since
S[x] = dt -mi;2
,
J ti
the Euler-Lagrange equations give (see Eq. (1.28))
This gives as solutions
Xcl(t) = v = constant. (2.50)
Thus, for the classical trajectory, we have
S[xd] = f f
dt ^mx = ^mv2
(tf - U). (2.51)
On the other hand, since v is a constant, we can write
Xf - Xi = V(tf - ti)
or, v=X
f^. (2.52)
Substituting this back into Eq. (2.51), we obtain
*M=Mt^)'<''-«-=^. (2
-53)
We recognize, therefore, that we can also write the quantum transi-
tion amplitude, in this case, simply as

This is a particular characteristic of some quantum systems which
can be exactly solved. Namely, for these systems, the transition
amplitude can be written in the form
U(tf,xf;ti,xi) = Ae^x
^, (2.55)
where A is a constant.
Finally, let us note from the explicit form of the transition ampli-
tude in Eq. (2.47) that
dU _ U im /xf — Xi
~dff ~ ~2{tf -U) ~~2htf-Uj
dU _ im fxf — %i TT
2 / „ 2
dxf h tf — t
d2
U _ im U (im fxf — Xi .
dxi h tf — ti h J tf — t
- J2
^ ( h U
4- — fXf
~Xi
2
Tf]
- V in
2{tf-U)+
2 t f - u ) )
2m {.BU . „ „ .
Therefore, it follows that
..8U ti2
d2
U , „ „ .
xh
wr-^!*f> (
2
-5
7
)
which is equivalent to saying that the transition amplitude obtained
from Feynman's path integral, indeed, solves the Schrodinger equa-
tion for a free particle (compare with Eq. (1.35)).

2.6 References
Das, A., "Lectures on Quantum Mechanics", Hindustan Book
Agency.
Feynman, R. P. and A. R. Hibbs, "Quantum Mechanics and
Path Integrals", McGraw-Hill Publishing.
Sakita, B., "Quantum Theory of Many Variable Systems and
Fields", World Scientific Publishing.
Schulman, L. S., "Techniques and Applications of Path Integra-
tion" , John Wiley Publishing.

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United States to Gibraltar. *** All the rest of the million
tons of shipping which crossed from the United States to
Gibraltar went across as single ships, going "on their own,"
as it were. These ships depended on their armed guard gun
crews, and were independent of the convoy system. They
actually encountered submarines, but they relied on their
guns for protection.
The convoy system, however, accomplished all that was expected of
it, and was markedly successful.
It was our destroyers at Queenstown, our forces on the French coast
and at Gibraltar, our cruisers escorting convoys crossing the Atlantic,
that made it the success it was—and it was one of the most
successful measures of the war.
President Wilson, as I have said, favored its adoption from the
beginning; in fact, wondered why the Allies had not adopted it upon
the outbreak of war in Europe. It was one of the first measures
recommended by the General Board. But at the time this country
entered the war, the Allies were pursuing exactly the opposite
method; that is, dispersion of shipping.
When troop transportation was first determined upon, in May, 1917,
we adopted the convoy system for troop-ships. It was in that month
that the British decided to try out the plan for merchant ships, to see
whether it would work. The first experimental convoy arrived in
England from Gibraltar, May 20. A few convoys were despatched in
June, and on June 22 Sims cabled me: "The British Admiralty have
now adopted the convoy system and will put it into effect as fast as
ships can be obtained for high sea convoy against raiders, and
destroyers for escort duty in submarine zone." He reported two
routes in operation, stated that eight convoys a week were planned,
and recommended that we furnish one cruiser or battleship a week
for high sea escort. On June 30, I informed him that the Department
would assign seven cruisers for this duty. Our destroyers were

engaged in the danger-zone from the time the first trans-Atlantic
convoys were started.
Putting the convoy system into effect was a big job, involving the
larger part of the world's shipping—a reversal of method that
necessitated a radical change in the naval scheme. Concerning the
part the United States Navy played in this great task, Admiral Sims
wrote in the World's Work:
I do not wish to say that the convoy would not have been
established had we not sent the destroyers for that
purpose, yet I do not see how it could have been
established in any complete and systematic way at such an
early date. And we furnished other ships than destroyers,
for, besides providing what I have called the modern
convoy—protecting the compact mass of vessels from
submarines—it was necessary also to furnish escorts after
the old Napoleonic plan. It was the business of the
destroyers to conduct merchantmen only through the
submarine zone. They did not take them the whole distance
across the ocean, for there was little danger of submarine
attack until the ships reached the infested waters. This
would have been impossible in any case with the limited
number of destroyers.
But, from the time the convoys left the home port, say New
York or Hampton Roads, there was the possibility of the
same kind of attack as that to which convoys were
subjected in Nelsonian days—that is, from raiders or
cruisers. We always feared that German cruisers or raiders
of the Moewe type might escape into the ocean and attack
these merchant ships, and we therefore had to escort them
across the ocean with battleships and cruisers just as they
did a century ago. The British did not have ships enough
available for this purpose, and here again the American
Navy was able to supply the lack; for we had a number of

pre-dreadnaughts and cruisers that were ideally adapted to
this kind of work.
AT GIBRALTAR, KEY TO THE MEDITERRANEAN
Above: U. S. S. Buffalo, Schley and Jupiter.

Inset: Rear Admiral Albert P. Niblack, commanding American naval
forces in the Mediterranean.
Below: The signal tower and American sub-chasers.

THE GREAT MINE BARRAGE AGAINST THE SUBMARINES
This map shows the location of the mine barrage across the North
Sea as well as the smaller one across the English Channel. The
dangers of this barrage, more than any other single factor, destroyed
the morale of the German submarine crews.

CHAPTER XII
SHUTTING UP THE HORNETS IN
THEIR NESTS
MINE BARRAGE ACROSS NORTH SEA A TERROR TO U-
BOATS—GERMANS PLANNED BIG DRIVE, BUT SUBMARINE
CREWS REFUSED TO GO TO SEA—MORALE SHATTERED,
KAISER'S NAVY WAS WRECKED BY MUTINY—PROPOSED
BY U. S. NAVY IN APRIL, 1917, EIGHTY PER CENT OF
BARRAGE WAS LAID BY AMERICANS.
Germany planned a great naval offensive in the fall of 1918—that is,
the German authorities did, the High Command. Why was it never
carried out? Why were the U-boats recalled? Why did the Kaiser's
High Seas Fleet surrender without striking a blow?
When Sir Eric Geddes, First Lord of the British Admiralty, visited
Washington in October, 1918, he told me that we might expect a
decided increase in submarine activity, a German drive at sea. In the
official conferences we held, Sir Eric and his associates predicted
that, notwithstanding all the efforts we were making, vastly more
tonnage might be sunk in the ensuing months. The British were

striving to increase ship production, and put as many war vessels as
possible into commission.
The next day I telegraphed the leading shipbuilders of the country,
asking them to come to Washington. Over 200 destroyers were
under construction or contracted for, and rapid progress was being
made on them. But I thought that, by special effort, we might rush a
larger number to completion. The critical situation outlined by the
British authorities was explained to the builders, and they were
directed to make construction continuous—to run three shifts of
eight hours each, working day, night and Sundays, and to speed up
to the utmost on destroyers and all anti-submarine craft. They
pledged their earnest assistance, proposing to increase forces, if
labor could be secured, and to push the program already undertaken
on the highest gear.
While the visit of the British mission as announced was to "discuss
certain matters concerning the naval situation," and its conferences
were confidential, its members in public statements made clear their
belief that easy or early victory was not to be expected.
"I have made it the keynote of all my policy and all my advice to
others not to be deluded with hopes of an early peace, but to
prepare for an ever-receding duration of the war," said Sir Eric
Geddes. "We must always be prepared for two years more, and then
only shall we have the sure means of victory in our hands."
More significant still, more to the point, was the remark made by Sir
Eric just before he sailed for Europe:
"A great renewed effort on Germany's part is impending. We know
it, and its extent."
Before he reached England, U-boat warfare was practically ended.
Within ten days the submarines were recalled to their home bases.
As they were returning to Germany they sank a few ships. But these
were the last few examples of German frightfulness on the seas.

What had brought about that tremendous change? It was not due to
any lack of determination on the part of the German Admiralty, or
the Kaiser. But they found that the big stick with which they were to
strike was only a broken reed. The morale of their navy was
shattered. Officers were willing enough to obey orders, but their
men refused to fight.
The U-boat crews, for years the pick and pride of the service,
refused to go to sea. Germany was building hundreds of submarines,
they were being turned out by the score. She might soon have sent
out a dozen for every one she had when ruthless warfare began. But
willing crews were lacking to man them.
This was a complete reversal of previous experience. A year before
U-boat duty had been the most sought-for branch of the service.
Essaying long voyages in the Atlantic or the Mediterranean, cruising
for weeks around the waters of England and France, their officers
and men had braved many dangers, and returning were hailed by
their countrymen as conquering heroes.
Sinkings had been made more difficult by the convoy system.
Listening devices had made it more dangerous for submarines to
remain in the vicinity of naval vessels. Patrol, by surface ships and
aircraft, had become more efficient. Shipping was more difficult to
get at and destroy. More submarines were being sunk than in the
early days. But, with all these operating against them, the U-boats,
even if they could not make such high scores in tonnage, had more
than an even chance to reach their home bases unscathed.
Now was another danger to face, however; one that was hidden and
deadly, and it had to be faced by every boat departing or returning.
Some U-boats, putting out to sea from their nesting places on the
German coasts, vanished utterly. No trace was left, no record of
what fate befell them.
Others, badly damaged, limped back to port. Survivors told of
colliding with mines hidden far below the surface, whose presence

could not be guessed. No vigilance could locate or action avoid
them. They might run into them anywhere within hundreds of miles.
This was a terror the undersea boatmen were unwilling to face. The
revolt of the U-boat crews spread to other branches of the naval
service, and the entire German navy began to disintegrate.
The mutiny in the German sea forces, the demoralization of its
personnel, has no parallel in naval history. This was undoubtedly due
to various causes, but, in my belief, there was no one thing that had
more influence in breaking the German morale, particularly in the U-
boat service, than did the Northern Mine Barrage.
Stretching across the North Sea, from Norway almost to the
Orkneys, this heavy barrier of powerful mines opposed any enemy
vessels which attempted to make their way around the north of
Scotland into the Atlantic. The Germans had only two exits from the
North Sea, the one covered by this mine barrier, and, to the south,
the narrow Straits of Dover, also partially mined and guarded by the
famous Dover Patrol.
It was a new factor in war, this vast barrage, the most successful
innovation, the biggest new naval offensive put forth after our
entrance into the war. American in conception, it was also mainly
American in construction. A joint British and American undertaking,
as it was, four-fifths of the mines laid were of American design and
manufacture, made in this country, taken across the Atlantic in
American ships, and laid by American naval vessels.
Though not actually laid until the summer of 1918, this was the first
big project proposed by the United States Navy after our entrance
into the war. In fact, it was only nine days after war was declared
that the Bureau of Ordnance presented an elaborate memorandum,
outlining the proposition. But the British Admiralty, without whose
consent and coöperation it could not be constructed, and Admiral
Sims pronounced it "impractical" and "unfeasible." It was not until
six months later that we secured the Admiralty's approval, and the
great project got under way.

The first mine was laid on June 8, 1918. "The barrier began to take
toll of the enemy's submarines as early as July 9, when one was
disabled on the barrier and compelled to return to Germany,"
reported Admiral Joseph Strauss, commander of American mining
operations in the North Sea. "It is not known how many submarines
were sunk or disabled in the mine field. It has been placed as high
as twenty-three. My own estimate, based on known sinkings, is ten,
although I am inclined to think that is a modest one."
Captain Reginald R. Belknap, commander of Mine Squadron 1, says
the barrage began to yield results before it was half way across.
"From the nature of the case it may never be known definitely how
many actually did come to grief there," he said; "but the best
information gives a probable ten before the middle of October, with
a final total of seventeen or more. In addition to this toll, the
squadron should be given credit for two submarines lost in the field
of British mines laid by the Baltimore off the Irish coast."
Eight and one-half per cent of the total number of submarines lost
during the war were brought into the list of missing by the barrage,
was the estimate of Admiral Ralph Earle, Chief of the Bureau of
Ordnance, under whose administration and leadership the mine
barrage was conceived, projected and constructed. Admiral Earle
reported to me:
It has been established that six submarines were lost in the
barrage and three more so badly damaged that they never
again put to sea. However, from further evidence, the
British Admiralty officially credit the barrage with fourteen
additional, or a total of twenty-three. Two hundred German
U-boats were destroyed in the war, or fifty more than the
Allies could account for. To err on the conservative side, we
claim but eight out of the fourteen credited the barrage by
the British Admiralty, or a total of seventeen. This is also
the figure arrived at by Captain R. R. Belknap, commander
of Mine Squadron 1. What does this figure show? Eight and

one-half per cent of the total number of submarines lost
during the war were brought into the list of missing by the
barrage, which existed for only six per cent of the period of
the war. Such results more than justified the effort and time
and funds expended.
PLANTING MINES IN THE NORTH SEA
A squadron of American mine planters at work. Inset: Rear Admiral
Joseph Strauss, who was in general command of mining operations.

HOW THE BIG MINES IN THE NORTH SEA BARRAGE WORKED
Fig. 1. Mine and Anchor leaving launching rails.
Fig. 2. Fifth wheel released, plummet supported by dashpot.
Fig. 3. Plummet released and unwinding cord.
Fig. 4. Plummet at end of cord. Slip hook pulled off.
Fig. 5. Anchor paying out mooring cable as it sinks. Plummet strikes
bottom and locks cable drum.
Fig. 6. Anchor on bottom submerging mine distance equal to length
of plummet cord.
A. MINE CASE
B. ANCHOR
C. PLUMMET
D. PLUMMET CORD
E. SLIP HOOK
F. MOORING
The barrage did more than take toll of submarines sent to kingdom
come by its mines. "There is no doubt," reported Sims in the
"Summary of Activities of American Forces in European Waters,"

"that the barrage had a considerable moral effect on the German
naval crews, for it is known that several submarines hesitated some
time before crossing. Also, reports from German sources are that the
barrage caused no small amount of panic in some of the submarine
flotillas. It is also probable that the barrage played a part in
preventing raids on Allied commerce by fast enemy cruisers."
Admiral Strauss, in his testimony before the Senate Investigating
Committee, declared that if the Northern Barrage and that across
the Straits of Dover had been fully completed as we planned, "it
would have ended the submarine menace, so far as submarines
going from the North Sea into the Atlantic were concerned;" and
that the building of the mine barriers across the Adriatic and Aegean
seas, for which we were preparing materials, "would have actually
ended submarine operations."
Could it have been built in 1917, a year earlier than it was? Strauss
said it could, and this was the firm belief of Earle and other
ordnance experts. True, the antenna mine we developed later was a
big improvement, superior to any previously devised. It would have
taken two or three times as many mines of the type then in use,
perhaps 180,000 of them, as was estimated. We manufactured
100,000 of the antenna type, and could have made as many more, if
necessary. The British had no antenna mines, Admiral Strauss
pointed out, and all the mines they laid in the barrage were of the
older type. After all the objections were presented to him, Admiral
Strauss, when asked if he still considered it would have been feasible
to have gone ahead with the barrage in 1917, unhesitatingly
answered: "Yes."
Not laying that barrage earlier—in fact, at the earliest possible
moment—was, in my opinion, the greatest naval error of the war. If
the British had erected it early in the war, and put a similar effective
barrier across the Straits of Dover and Otranto, the Germans would
have been so restricted that widespread U-boat warfare, with its
terrible destruction of life and shipping, would have been impossible.

"Shutting up the hornets in their nests," as President Wilson
expressed it, was the first idea that occurred to us when we went to
war. The Bureau of Ordnance on April 15, 1917, submitted a
memorandum urging that we "stop the submarines at their source"
and suggesting that mine barriers be laid across the North Sea, the
Adriatic and the Dardanelles. "The northern barrier," it stated,
"would extend from the mid-eastern coast of Scotland to the
Norwegian coast, a distance of about 250 miles," and the southern
(that is, to close the Straits of Dover) would extend "from the
southeast coast of England and to a point on the French coast near
the Belgian frontier, a distance of about forty miles." Next day I
cabled Admiral Sims, who had just arrived in London:
Is it not practicable to blockade German coast efficiently
and completely, thus making practically impossible the
egress and ingress of submarines? The steps attempted or
accomplished in this direction are to be reported at once.
Two days later came the answer:
To absolutely blockade the German and Belgian coast
against the entrance and departure of submarines has been
found quite unfeasible.
The next day he wrote a long letter, amplifying the difficulties and
reporting against any such barriers. But our ordnance experts were
thoroughly convinced the project was feasible. On May 9th they
outlined their plans in a memorandum to be submitted to the British
Admiralty, and on May 11th I cabled to Admiral Sims: "Much opinion
is in favor of concerted efforts by the Allies to establish a complete
barrier across the North Sea, Scotland to Norway, either direct or via
the Shetlands, to prevent the egress of German submarines." I
added, "The difficulty and size of the problem is recognized, but if it

is possible of accomplishment the situation would warrant the
effort." He was directed to consult with the British Admiralty
regarding this plan. Two days later came the reply:
From all experience Admiralty considers project of
attempting to close exit to North Sea to enemy submarines
by the method suggested to be quite impracticable. Project
has previously been considered and abandoned.
In a dispatch on May 14th Sims said: "The abandonment of any
serious attempts at blockading such passages as Scotland-Norway,
the Skagerrack and Scotland to Shetland has been forced by bitter
and expensive experience."
"As may well be imagined," he wrote later, "this whole subject has
been given the most earnest consideration, as it is, of course,
realized that if submarines could be kept from coming out, the whole
problem would at once be solved." But he said, "I cannot too
strongly emphasize the fact that during nearly three years of active
warfare this whole question had been the most serious subject of
consideration by the British Admiralty," which had concluded that no
"barrier can be completely effective."
This, however, did not deter our ordnance experts. The more they
studied the question, the more were they convinced that the barrier
could be "put across." Believing in mines, preparing for mine
operations on a large scale, they were astonished when, on May
31st, Sims reported that, instead of our giving attention to mine
production, the British Admiralty "consider we can more profitably
concentrate on other work."
Earle and his associates in the Bureau of Ordnance never doubted
final success. They experimented with mines, firing and anchoring
devices, and on July 30th announced the development of a new type
of mine, particularly adapted to deep waters. A unique feature of

this mine was that it did not have to be struck to explode, but would
explode if a submarine passed close to it. This was due to the firing
apparatus, which was evolved from an electrical device submitted by
Mr. Ralph C. Browne, of Salem, Mass., to be used on a submerged
gun. Officers of the Bureau concluded this could be adapted to
mines, and in May began work to that end. Commander S. P.
Fullinwider, chief of the Mine Section, was aided by Mr. Browne,
Lieutenant Commander T. S. Wilkinson, Jr., and Commodore S. J.
Brown in producing this firing device, and others who assisted in
developing the mine were Lieutenant Commanders O. W. Bagby, J.
A. Schofield, W. A. Corley, C. H. Wright and H. E. Fischer, Lieutenant
S. W. Cook and Lieutenant (junior grade) B. W. Grimes.
With this improved mine as an argument, our ordnance officers
renewed the proposal of a mine offensive in the North Sea. The
memorandum the Bureau submitted was comprehensive, and
contained all the essential features of the barrage plan that was later
adopted and carried into effect.
How could the project be best presented to the British Admiralty
again? Admiral Mayo was preparing to sail within a short time for
Europe. Just before his departure the entire project was discussed
and the operation of the improved mines explained, as he was to
bring the whole matter to the attention of the British Admiralty and
the Allied Naval Council. To prevent loss of time and further insure
the Admiralty's consideration, on August 17th, before Mayo sailed, I
cabled Sims:
Bureau of Ordnance has developed a mine which it hopes
may have decisive influence upon operations against
submarines. Utmost secrecy considered necessary. Request
that an officer representing the Admiralty, clothed with
power to decide, be sent here to inspect and thoroughly
test mine, and, if found satisfactory, arrange for
coöperation in mining operations.

The Allied Naval Conference, held in London September 4th and 5th,
which Mayo attended, took up not only the barrage project, but
another proposition our Navy Department had suggested months
before, a close offensive in German waters. After the meeting Mayo
cabled:
Conference completed after agreement upon the following
points:
1. That close offensive in German waters should be
carefully considered by Allies, after which they should
indicate to British Admiralty contribution of old war ships
they are prepared to furnish should offensive prove
practicable.
2. That alternative offensive employing effective mine
field or mine net barrage to completely shut in North Sea
not practicable until adequate supply satisfactory type
mines assured, and that pending such supply, extension
present system mine fields desirable and that mine net
barrage impracticable.
This indicated to us that the British still doubted the effectiveness of
a barrage, as well as our ability to furnish an adequate supply of
mines. It was evident that, after five months of earnest advocacy,
further urging was needed to secure approval of the project.
Benson, therefore, on September 12th, cabled Mayo:
There are great possibilities in the satisfactory solution of
the mine and depth-charge question. Officers sent over
here most satisfactory and remarkably well posted. I think
it would help the whole situation wonderfully if Commodore
Gaunt could visit the Admiralty for a few days and have a
heart-to-heart talk. No time to be lost.

What happened next? The day after Benson's message was
received, the British Admiralty made out for Mayo a paper entitled,
"General Future Policy, Including Mine Policy," with an appendix,
"Mine Barrage Across the North Sea." The policy outlined by the
Admiralty, announced September 14th, was the same the Navy
Department had suggested nearly five months previous.
Even then there was delay. On October 9th, Sims reported that the
Admiralty was "thoroughly investigating the question" and that "the
discussion of this question will probably be postponed by the
Admiralty until the return of the commander-in-chief." We were still
not certain as to whether the British were ready to put it through.
But, believing that the plan must finally be put into effect, our
Bureau of Ordnance went ahead, and let the contracts for 100,000
mines. Upon Mayo's return October 15th, the amendments
suggested by the British were approved by the General Board and
accepted by the Department. Nothing definite, however, had come
from London and on October 20th Sims was cabled:
The Department requests to be informed whether the plan
for the placing of a mine barrier across the North Sea on
the Aberdeen-Egersund line has the approval of the
Admiralty.
Finally on October 22nd, an answer direct from the British Admiralty
said, "Admiralty has approved mine barrier and now confirms
approval."
All the details were then perfected—this required several days—and
on October 29th I received and approved the completed plans. The
President, who for months had been impatient of delay, gave his
approval as soon as they were laid before him. This was at a cabinet
meeting on October 30th. The same day a cable was sent to the
Admiralty that we had taken steps to fit out mine-planters; that

shipment of mines would begin the first of January, and officers
would be sent in a few days to arrange details.
So after months of opposition, doubt and indecision, the two navies
united in the construction of this most stupendous job of the kind
ever conceived or undertaken. It was well done and the result
demonstrated its effectiveness. Admiral Sims himself, after its
completion and success, said that "no such project has ever been
carried out more successfully" and that "as an achievement it stands
as one of the wonders of the war."
I am not giving these details in any spirit of criticism of the British
Admiralty or our representative in London, but to do justice to the
vision, initiative and resource of the American Navy. It was, indeed,
a bold and gigantic experiment, calling for many millions of money
and the strenuous and dangerous work of many men. That it was so
successfully done reflects credit alike on Britons and Americans, and
both share in the honor of its accomplishment.
Manufacturing 100,000 mines was a big order, but that was only the
beginning. They had to be shipped 3,500 miles overseas, which
necessitated a fleet of mine-carriers. Twenty-three cargo vessels
were converted, and assigned to this duty. To fill the mines with
explosives a mine-loading plant of 22 buildings was erected at St.
Julien's Creek, Va., capable of receiving, loading and shipping 1,000
mines a day. Advanced bases, for inspection and assembly of the
mines, were established in February, 1918, on the east coast of
Scotland, at Inverness and Invergordon, with Captain O. G. Murfin in
charge.
For the work of mine-laying, a Mine Squadron was created, under
command of Captain Reginald R. Belknap. This consisted of the
flagship San Francisco (Captain H. V. Butler), and her consort, the
Baltimore (Captain A. W. Marshall), "crack cruisers of the vintage of
1890," as Captain Belknap called them; and eight former merchant
vessels converted into naval mine planters. Four of these were
Southern Pacific or Morgan liners, carrying freight between New York

and Galveston, renamed the Roanoke (Captain C. D. Stearns),
Canonicus (Captain T. L. Johnson), Housatonic (Captain J. W.
Greenslade), and Canandaigua (Commander W. H. Reynolds). Two
were the Old Dominion passenger liners Jefferson and Hamilton,
running between New York and Norfolk, renamed Quinnebaug
(Commander D. Pratt Mannix), and Saranac (Captain Sinclair
Gannon). The remaining two were the fast Boston and New York
passenger steamers, Massachusetts and Bunker Hill, of the Eastern
Steamship Corporation, renamed Shawmut (Captain W. T. Cluverius),
and Aroostook (Captain J. Harvey Tomb). They were accompanied
abroad by several seagoing tugs, the Sonoma, Ontario, Patapsco and
Patuxent.
Admiral Strauss, who was in general command of mining operations,
went to England in March, inspected the bases, and conferred with
the British authorities as to the general arrangements. His flagship
was the Black Hawk (Captain R. C. Bulmer), which was also the
repair vessel of the mine force. The British began mine laying in
March, but one of their vessels, the Gailardia, was sunk; and
operations were suspended for a time until the safety of the mines
could be assured.
The Baltimore, the first of our vessels sent over, arrived in the Clyde
in March. Submarines were very active in Irish waters, and the
Admiralty decided to lay a deep mine-field off the north coast of
Ireland, in the North Channel. As all British mine-layers were
employed elsewhere, the Admiralty requested the use of the
Baltimore. This was readily granted and the Baltimore engaged in
this from April 13th until the latter part of May, joining our squadron
in Scotland June 2nd. The Roanoke, sent over to assist her, was
instead ordered to our base at Invergordon.
Sailing from Newport, May 12th, the San Francisco and other vessels
arrived at Inverness, May 26th, all ready to begin operations. Twelve
days later the squadron started on its first mine-planting "excursion."
On these expeditions, which lasted usually from 40 to 80 hours, the
squadron was regarded as a part of the British Grand Fleet.

Screening it against submarines, and hostile mines casually placed,
was an escort of eight to twelve British destroyers, which formed
around the squadron upon its leaving the base and kept with it until
its return. To guard against attack from enemy cruisers, while away
from the coast, the squadron was accompanied by a supporting
force, consisting of a battleship or battle-cruiser squadron and a
light-cruiser squadron of the Grand Fleet, sometimes by all three,
according to the estimated probabilities of attack. On the second
mining excursion the support was the Sixth Battle Squadron, the
American battleships, commanded by Admiral Rodman. Captain
Belknap gave a vivid picture of the dangerous character of mine-
laying when he said:
One may imagine with what feelings we saw our own great
ships file out of Scapa Flow, form line on our quarter, and
slowly disappear in the haze, as they swept off to the
southeastward. It will be readily understood that the way
had to be made smooth for the mine planters. As long as it
was so, all would go well; but a single well placed torpedo
or mine, or a few enemy shells, would certainly finish one
vessel, and probably destroy all ten of them. Each mine
planter carried from 24 to 120 tons of high explosive, a
total of nearly 800 tons in the squadron, many times more
than the amount that devastated Halifax. With this on
board, the squadron was hardly a welcome visitor
anywhere.
Operations as a whole were conducted in conjunction with a British
mine-laying squadron of four vessels, under command of Rear
Admiral Clinton-Baker. American and British squadrons often went
out at the same time, under protection of the same heavy vessels,
but except on two occasions they worked separately, in different
parts of the barrage area. Thus there were altogether fourteen mine
planters at work at the same time.

On the first excursion, June 7th, the American squadron planted a
mine field 47 miles long, containing 3,400 mines, in 3 hours and 36
minutes. Everything went without a hitch. One ship emptied herself
of 675 mines without a single break, one mine every 11-1/2 seconds
through more than two hours, a record never before equalled.
ONE OF THE PERILS OF MINE-SWEEPING
An explosion close astern of the Patapsco. The greatest care was
exercised to avoid accidents of this character, but to eliminate them
entirely was impossible.

THE MINE-SWEEPERS PROVED WONDERFUL SEA BOATS
These tiny craft rode many a rough sea which worried larger and
more powerful ships.

UNITED STATES NAVAL OFFICES IN IMPORTANT COMMANDS
Left to right: Admiral Sims, Admiral Mayo, Captain Nathan C.
Twining, Captain O. P. Jackson, Admiral Wilson.

AMERICAN AND BRITISH NAVAL OFFICIALS
Left to right: Admiral Benson, Secretary Daniels, Sir Eric Geddes,
Admiral Duff.
Dangerous as was the work, there were very few casualties. One
man fell overboard from the Saranac and was drowned, but he was
the only man lost at sea, and there were but four other deaths in
that force of 4,000. Laden with high explosives, navigating waters
where enemy mines had been laid, operating near mine fields, and
in danger of premature explosion from those they themselves had
laid, it is remarkable that not one of these ships was lost or seriously
damaged.

The eighth excursion in which British and American squadrons
joined, both in command of Admiral Strauss, closed the western end
of the barrier, off the Orkneys. The next expedition was conducted in
the same manner, with Rear Admiral Clinton-Baker, of the British
Navy, in command. The American squadron made fifteen excursions,
the British eleven, operations being completed October 26th. In four
hours on one expedition, 6,820 mines were planted, 5,520 by our
vessels, 1,300 by the British. Our squadron alone planted a field 73
miles long in one day.
Seventy thousand, two hundred and sixty-three mines were laid—
13,652 British, 56,611 American. Numerous lines were laid near the
surface; others were placed at from 90 to 160 feet; and the lowest
went to depths from 160 to 240 feet.
Beginning near the northern Orkneys, the barrier ran to Udsire Light,
near Bergen, on the coast of Norway, 230 miles. Its average width
was 25 miles, in some places it was 35 miles across, and at no point
was it less than 15 miles wide. At its narrowest, this meant more
than an hour's run for a submarine. Mines were planted, row after
row, at various depths. If a U-boat proceeded on or near the
surface, it would encounter from six to ten lines of mines. If it tried
to break through by going deeper, there were more of the deadly
explosives. Submergence was, in fact, as dangerous as running the
gauntlet on the surface. No matter how far the sub went down there
were mines to meet it, to the furthest limit of submarine descent.
One touch—even a slight jar from the vibration of the U-boat—was
enough to set off one of these mines, and when it exploded the U-
boat was done for.
Mine-laying was not the only role played by the American force,
Captain Belknap wrote:
In addition to the value of the barrage itself, in keeping the
enemy submarines in or from their bases, the mine
squadrons were expected to serve as bait, to draw out the

German fleet; the squadrons' role being neatly expressed
by one high officer as "an important military offensive with
a front seat at the Second Battle of Jutland." This ever
present possibility and the fact that the working ground lay
in the principal thoroughfare of enemy submarines, with
attendant incidents of periscope sightings, submarine
reports, depth charges, smoke screens, floating mines, and
dead Germans floating by, lent spice to the work, which,
like the proverbial sporting life, was often hard but never
dull. ***
On every excursion, during the mine laying, one or more of
the mines would go off fairly close astern—lest we forget!
The mines were very sensitive, and no witness of an
excursion could retain any doubt as to the fate of a
submarine that "luckless dares our silent wake."
The eastern end of the barrage extended to the territorial waters of
Norway. That country being neutral we could not, of course, mine to
its shores. With the growth of the barrier, U-boats took advantage of
this, going within the three-mile limit to slip by into the open sea.
The Norwegian Government then announced its decision to mine its
waters, which closed that gap.
Our original plan was to plant mines clear to the Orkneys, and this
we urged. But Admiral Beatty and others strongly objected, fearing
that it might hamper the operations of the Grand Fleet. So the mine-
fields ended ten miles east of the islands. But this ten-mile passage
was heavily patroled, and any "sub" attempting to pass that way
must run the risk of attack by numerous naval vessels. Thus the U-
boats could not get through anywhere except at great risk. Months
were required to lay that barrier, and during that time there were
unmined areas through which vessels could pass.
The barrage was completed October 26th, almost coincident with
Germany's recall of its U-boats, which practically ended submarine

warfare. Some of those recalled did not reach these waters until the
armistice had been signed, hostilities were over, and they were
immune from attack. Some "ran" the barrage, and several met the
fate of the U-156, one of the undersea cruisers which operated off
our own coasts. Attempting to get through the barrier, she struck a
mine and went down. So far as known, only 21 of her crew were
saved.
The Northern Barrage cost us approximately $80,000,000. Shipping
sunk by submarines averaged, for a long period, over $70,000,000 a
month, at times ran over $80,000,000, in actual monetary value, not
counting the resultant military effect of its loss. Admiral Sims
estimates that the war cost the Allies $100,000,000 a day. Thus, if
the Northern Barrage shortened the war one day, it more than
repaid its cost.
Our mining projects were not confined to the North Sea. Plans had
been accepted and mines were in process of manufacture for a like
barrage across the Straits of Otranto, from Brindisi, the heel of Italy,
to Saseno Island. This would have effectually shut up German and
Austrian submarines in the Adriatic. We had also agreed to
undertake to provide and lay 26,800 mines for a barrage in the
Aegean Sea from Euboea Island to Cape Kanaptitza, except for the
part resting on Turkish territorial waters, which was to be established
by Great Britain, since the United States was not at war with Turkey.
The armistice made these barrages unnecessary.
But our mining operations were by no means concluded with the
cessation of hostilities. Clearing the seas was our next duty, for
navigation would not be safe until the many thousands of mines
were removed. This work was divided among the various nations.
The United States volunteered to remove all the mines we had laid.
Admiral Strauss, in charge of these operations, had his base at
Kirkwall, and his force comprised 34 mine-sweepers, 24 sub-chasers,
two tugs, two tenders and 20 British trawlers, which were also
manned by U. S. naval personnel:

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Field theory a path integral approach 2nd Edition Ashok Das

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