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Path Integrals in Quantum Theories:

A Pedagogic First Step

As I have mentioned elsewhere (

Quantum Field Theory: A Pedagogic Intro

)

, I strongly

believe it far easier, and more meaningful, for students to learn QFT first by the canonical

quantization method, and once that has been digested, move on to the path integral (many paths)

approach. Hopefully, the material below will help such students, as well as those who are forced

to begin their study of QFT via path integrals.

1 Background Math: Examples and Definitions

1.1 Functionals

Functionals form the mathematical roots of Feynman’s many paths approach to quantum

theories. A functional is simply a function of another function.
Example: Kinetic energy

where

(

is a function of

, and

is a function of

Definition(s): A functional is 1) a function of a function, OR equivalently, 2) a function of a

dependent variable, OR equivalently, 3) a mapping of a function to a number.
Symbolism:

[ ]

[

]

( ) or

( ), ( )

F x t

F x t x t

(1)

The square bracket notation is common, but not always used. Mathematically,

and

represent any function and its independent variable, though in physical problems, they are

typically spatial position and time, respectively. Functionals are often dependent on the

derivative(s) of a function, as well as the function itself, as in the RHS of (1). Total energy, with

potential energy dependent on

(

) and kinetic energy dependent on ( )

x t

, is one example.

Additionally, a functional could also be a direct (rather than indirect as in (1)) function of

, i.e.

[

]

( ), ( ),

F x t x t t

(2)

1.2 Functional Derivative

Definition: A functional derivative is simply the derivative of a functional (

above) with respect

to a function upon which it depends (

above).

Symbolism:

( )

x t

(3)

The

notation is common, though the partial derivative symbol is often used instead.

1.3 Functional Integral

Definition: A functional integral is the integral of a functional with respect to a function upon

which it depends.
Symbolism:

( )

F x

F x t

(4)

In the literature, one may find use of the usual differential symbol

instead of

2 Different Kinds of Integration with Functionals

The value of a functional

of a physical system, such as a particle, is dependent on where it

is in space and time, i.e.,

(

) and

in (1) are then considered spatial position and time. Further,

one can integrate a functional

in different ways over its path in space and time, or over

projections of that path. Several of these are depicted in Table 1 below. The first three kinds of

integration shown below are fairly self explanatory. We comment on the fourth after the table.

Table 1. Some Ways to Integrate Functionals

Type of

Integration

Graphically

Math

Comment &

Use in Physics

Area over the

path in

x(t)

vs.

space

Fds

where

distance

along path

No real physical

application.

Projection of

the area in 1

onto the

F-t

plane

Fdt

F=L

, the Lagrangian,

then this integral =

the action.
Classically,

minimum (or

stationary) for physical

paths

Projection of

the area in 1

onto the

F-

(

) plane

( )

F x t

This is the usual

definition of

“functional integral”
This is starting point for

4, below

Simultaneous

integration

over all

possible paths

in 3

( )

F x t

QM & QFT Feynman

path integral approach.

symbol implies a

sum of the integrals of

all paths in 3, not just

the classical path

x(t)

[

x(t)

]

path in

x-t

space

along path

x(t)

[

x(t)

]

projection onto

F-t

plane

x(t)

[

x(t)

]

projection onto

F-x

plane

x(t)

[

x(t)

]

4 of an infinite

number of paths

The fourth way to integrate above is not simple, nor is its purpose at all obvious at this point.

We devote entire sections below to explaining its origin, its value, and means to evaluate it. So,

for now, just let it float easily through your head and don’t bother straining to understand it.
Alternative nomenclature: Because functional integrals are integrated over particular paths (in

space in above examples), they are often also referred to as path integrals.

3 The Transition Amplitude

3.1 General Wave Functions (States)

Recall from QM wave mechanics, that for a general normalized wave function

equal to a

superposition of energy eigenfunction waves (which are each also normalized),

1 1

3 3

(5)

is the amplitude of

, so the probability of finding

upon measuring is

1 1

A A

(6)

If we were to start with

initially, and measure

later, the wave function would have

collapsed, i.e., underwent a transition to a new state. (6) would be the transition probability.
Definition: The transition amplitude is that complex number, the square of the absolute

magnitude of which is the probability of measuring a transition from a given initial state to a

specific final state.
Symbolism: The transition amplitude is often written as

( ,

; )

ψ ψ

(7)

implying an initial state

, a final state

, and an elapsed time between measurements of the

two of

This terminology carries over to QFT when particles change types. For example, the

probability that an electron and a positron would annihilate to create two photons would be the
square of the absolute value of the transition amplitude between the initial (e

–

, e

) and final (2

)

states. (Almost all of QFT is devoted ultimately to determining the transition amplitudes for the

different possible interactions between particles.)
Schroedinger Approach Amplitudes

We can’t get into explaining it here (for those who may not know it already), but the

Schroedinger approach to QM leads to an expression of the transition amplitude of form

initalstate

final state

measured

evolved state

( ,

; )

iHT

T t

ψ ψ

−

(8)

where

is the Hamiltonian operator.

Alternative nomenclature: The transition amplitude

is sometimes called the propagator

(though

not

the “Feynman propagator” of QED) because it is the contribution to the wave

function at

at time

from that at

at time 0. It “propagates” the particle from

3.2 Position Eigenstates

When the particle has a definite position, e.g.,

, the wave function is an eigenstate of

position, and the ket is written |

. The transition amplitude for measuring a particle initially at

, and finally at

, would take the form

evolved state

( , ; )

iHT

U x x T

x e

−

(9)

In wave mechanics notation, |

and

are both delta functions of form

(

x-x

) and

(

x- x

), the first of which

is represented schematically on the

left in Figure 1. As the initial state
evolves into

, however, it, like wave

packets generally do, spreads, and its

peak diminishes (see wave function

envelope on right side of Figure 1.)

The amplitude for measuring the

particle at time

, i.e., for

measuring a delta function |

that

collapsed from

, is (9).

We can re-write (9), in wave mechanics notation as

( , ; )

(

) ( , )

( , )

U x x T

x x

x T dx

x T

+∞

−∞

−

(10)

Thus,

( , ; )

*( , ) ( , ) probability density at

U x x T

x T

(11)

Modification to definition: Hence, the square of the absolute value of the transition amplitude

for eigenstates of position is

probability density

not

probability

, as was the case for energy

eigenstate wave functions of form (5).

As we will see, the value found using the RHS of (9), i.e., that of the Schroedinger approach,

is the same as the value found using Feynman’s many paths approach.

4 Expressing the Wave Function Peak in Terms of the Lagrangian

4.1 Background

One of Feynman’s assumptions for his path integral approach to QM and QFT was to

express the wave function value at the peak of a wave packet (see Figure 1) in terms of the

Lagrangian (exact relation shown at the end of this section 4). I have never seen much

Dirac delta

function

Wave

function

envelope at

time

Wave

function

envelope at

time = 0

| |

Peak velocity =

Group velocity

Figure 1. Propagation of a Position

Eigenstate Quantum Wave

justification for this in the literature, other than it is simply an assumption that works (so learn to

live with it and move on!)

In the present section I have taken a different tack, by providing rational for why we could

expect Feynman’s form of the wave function peak to work. The logic herein may well parallel

what went on in Feynman’s mind as he was developing his path integral approach.

4.2 Deducing Feynman’s Phase Peak Relationship

4.2.1 The Simplified, Heuristic Argument

In QM, the plane wave function solution to the Schroedinger equation,

(

) /

i Et i

−

− ⋅

p x

(12)

means the phase angle, at any given

and

, is

(

) /

= −

− ⋅

p x

(13)

If we have a particle wave packet, it is an aggregate of many such waves, so it is not in an

energy or momentum eigenstate. However, it does have energy and momentum expectation

values that correspond to the classical values for the particle. The wave packet peak travels at

the wave packet group velocity, which corresponds to the classical particle velocity.

Now, imagine we approximate the wave packet with a (spatially short) wave function such

, where

and

take on the values of the wave packet expectation values for energy and

momentum, respectively. If

represents the position of the wave packet peak (the middle of our

approximated wave function

), the time rate of change of phase at the peak is then

(

)

T V

− − ⋅

− − + ⋅

p v

(14)

where

is the velocity of the wave peak,

is kinetic energy, and

is potential energy. Non-

relativistically,

(15)

so, in terms of the classical Lagrangian

T V

−

(16)

More formally, using the Legendre transformation

(

here)

i i

L p q H

−

= ⋅ −

p v

(17)

directly in (14), we get (16).

Thus, from (16), the phase difference between two events the particle traverses is

(18)

where

is the classical action of Hamilton. The classical path between two events is that for

which the Hamiltonian action is least. Note that (18) is an integral of type 2 in Table 1.

Hence, the wave function at the peak could be written in terms of the Lagrangian as

peak

(19)

This is the typical starting point assumption when teaching the Feynman path integral

approach (still to be developed beginning in Section 5.)

In relativistic quantum mechanics (RQM) and quantum field theory (QFT), we get a solution

form similar to (12) (differing only in the normalization factor

), and thus (14) is also true

relativistically. Further, since (17) is true relativistically, as well, then so are (16), (18), and (19).

4.2.2 More Precise Argument

The precise expression for a QM particle wave packet

, where overbars designate

expectation (classical) values;

, the group (peak, classical) velocity; and

(

), the momentum

space distribution is

(

)

(

)

(

)

(

)

1 for

peak,

time depend

real

i.e., for

&complex

( ) for

( , )

( )

peak

x v t

i t

p p

Et px

v t x p p

A t

x x

x t

g p e

+∞

−∞

−

(20)

We are interested in the value of (20) at the peak,

(

peak

), where

peak

. To begin,

note that with

x=x

peak

inside the integral, the exponent of the second factor in the integrand

equals zero, and so that factor equals one. The function

(

) is typically a real, Gaussian

distribution in

p p

−

, and independent of time. The third factor in the integrand is complex and

time dependent.

Thus, with

x=x

peak

, the integral in (20) is a function (generally complex) only of time, which,

along with the factor in front, we will designate as

(

). Thus, for the entire history of the wave

packet, the wave function value at the peak is

(

)

(

, )

( )

peak

i Et px

A t e

−

(21)

Except for the time dependence in

(

), this is equivalent to (12), as the expectation values

for

and

equal the classical values for the particle. So, with regard to the exponent factor in

(21), all of the logic from (13) through (19) applies here as well. The final result is so important,

we repeat it below, with

being the classical particle Lagrangian,

representing the time when

the peak is detected, and phase at

= 0 taken as zero. The RHS comes from (10).

(

)

( )

( , , )

peak

A T e

U x x T

(22)

We evaluate

(

) exactly in the Appendix.

Definition: Borrowing a term from electrical engineering, we will herein refer to

as a phasor.

5 Feynman’s Path Integral Approach: The Central Idea

Feynman’s remarkable idea takes a little getting used to. He reasoned that a particle/wave

(such as an electron) traveling a path (world line in spacetime) between two events could

actually be considered to be traveling along all possible paths (infinite in number) between those

events.

Difficult as it may be, initially, to believe, we will see below that the result from

superimposition of the phasors from all of these paths gives us the same result as if we used the

standard QM theory of Schroedinger with a single wave. The two different approaches are

equivalent.
Definition: Feynman’s method is called the “path integral”, “many paths”, or “sum over

histories” approach to QM (and QFT).

Note that the paths do not have to satisfy physical laws like conservation of energy,

least action, etc. Moreover, each possible path is considered equally probable.

We will lead into the formal mathematics of the many paths approach by first examining

simple situations with a finite number of paths between two events.

6 Superimposing a Finite Number of Paths

6.1 The Rotating Phasor

The phasor of (22) can be expressed in the complex plane as a unit length vector with angle

relative to the real positive (horizontal) axis. As time evolves this vector rotates at the rate

i.e., the total phase

. So we can picture the phasor

as a unit length vector rotating like a hand on a clock in a 2D

complex plane (though it is a counterclockwise rotation).

For the purposes of Feynman’s approach, we can consider

the particle as a wave packet with phase at the peak

determined by (22), and our final measurement a position

eigenstate measured at the packet peak. We then imagine a

different wave packet following each one of the infinitely

many paths between two specific events. We visualize the

phasor at the peak for each of these paths as a vector rotating in the complex plane as time passes

(i.e., as the wave packet peak moves along the path), eventually having a particular value at the

final event, the arrival place and time. Each path will have a different final phase.

6.2 Several Paths Graphically

Fig. 24 in Feynman’s book

QED: The Strange Theory of Light and Matter

, is an insightful,

somewhat heuristic, illustration of the many paths concept for light. Since we wish to focus, for

the time, on non-relativistic quanta, we employ a similar, and at least equally heuristic,

illustration in Fig. 3 for an electron rather than a photon. In Fig. 3 an electron is emitted at event

a, reflected, like light from a mirror, off of a scattering surface, and detected at point b. The

scattering surface might be difficult to construct in practice, but one can imagine a surface

densely packed with tightly bound negative charge.

We look at a representative 15 different paths for the electron, out the infinite number in the

many paths approach, and label them with letters A to O. Each path takes the same time

. Note

that path H is the classical path, having equal angles of incidence and reflection. Since it is the

shortest, particle speed for that path is lowest.

The Lagrangian here is simply the kinetic energy, and this is constant, though different, for

each path. Since speed is least for the classical path H, it has the smallest Lagrangian, and thus

i L/ dt

= L/ dt

Figure 2. Rotating Phasor

the least action. The other paths do not obey the

usual classical laws, such as least action, equal

angles of incidence and reflection, etc. But

according to Feynman’s approach, we have to

include all of them.

From (22) and Fig. 2, we can determine the

phasor

iS/

of (22) for the particle/wave arriving at

event b, for each path, where

LT =

The phasor direction in complex space for each

path at the detection event b is depicted in the

middle of Fig. 3.

The bottom part of Fig. 3 shows the addition

of the final event phasors for all 15 paths. Note

that the paths further from the classical path H tend

to cancel each other out, because they are out of

phase. Conversely, H and the paths close to H are

close to being in phase, and thus, reinforce each

other via constructive interference. So, the primary

contributions to the phasor sum are from those

paths close to the classical path.

If we were to increase the number of paths, the

jaggedness of the curve formed by the 15 phasors

would smooth out, but its basic overall shape

would remain essentially the same. If we were to

increase the Lagrangian, while keeping speed the

same for each path (i.e., increasing mass of the

particle), phasors now near the middle of the curve

would shift towards the ends, and thus, be

cancelled out via interference. In other words,

increasing mass brings us closer to the classical

case, and the paths closer to classical then make

greater contributions to the final sum. A similar effect would occur if the value for Planck’s
constant were smaller. As 0, all paths but H would tend to cancel out.

Footnote:

I used to think that increasing mass, and thus getting closer to the classical situation, would bring the

phase angle of the sum-of-all-paths phasor in directional alignment with H, the classical path phasor (or at least with

of (22).) However, this is not the case. The important thing in Feynman’s approach is not the phase of the sum-

of-all-paths phasor, but its

length

, which is proportional to |

|. And this length gets greater contribution from paths

nearer H than from those further away.

Note that in order to get a graphically significant Fig. 3, I had to use a value for almost eight orders of

magnitude greater than the actual value. Otherwise the phase angles between adjacent paths, for the relatively large

spacing between paths of the figure, would have resulted in a seemingly random hodgepodge of phasors, and

obscured, rather than

illumined, the real physics involved.

If you would like to experiment with changing values for mass, , and number of paths yourself, download the

Excel spreadsheet

Many Paths Graphic Electron Reflection

. End footnote.

Many Paths Electron Reflection

C D E F G H I

J K L M N O

electron

scattering
surface

electron
emission

electron
detection

Action

A B C D E F G H I

J K L M N O

Phasor direction of each path at event b

Phase Addition of All Paths at Event b

Figure 3. Graphical Justification for

Many Paths Approach

Feynman intuited that the amplitude of the final phasor sum was extremely meaningful.

That is, the square of its absolute value (i.e., the square of its length in complex space) was

proportional (approximately, for a finite number of paths; exactly, for an infinite number) to the

probability density for measuring the photon/particle at event b. What we mean by

“proportional” should become clearer after the following three sections.

6.3 Many Paths Mathematically

Consider particle paths similar to those of Fig. 3, where the wave function peak for path

number 1, with

(

) as in (22), as

( )

peak

A T e

(23)

In the spirit of the prior section, one considers the phasor of (23)

without

(

) as representing the

particle, AND that particle is considered to simultaneously travel many paths between events a

and b. Then, the summation of the final phasors for each path is expressed mathematically as

sum

.....

A e

(24)

where

is the amplitude of the sum. As the number of paths approaches infinity, |

becomes

proportional to the probability density of measuring the particular final state

at event b. That is,

sum

lim

( , , )

(probability density)

A e

U x x T

A e

→∞

∝

. (25)

We will learn how to evaluate the limit in (25).

6.4 Another Example

Consider a double slit experiment with a classical Huygen’s wave analysis showing

alternating fringes of light and dark, which via the classical interpretation is caused by

constructive and destructive interference of light/electron waves.

By the Schroedinger wave approach, a single quantum wave travels through both slits,

interferes with itself, either constructively or destructively, to result in a wave amplitude that

varies with location along the receiving screen. The probability density (square of the amplitude

absolute value) of finding a photon/electron also varies with that screen location. So as the

quantum waves collapse, one at a time, on the screen, they tend to collapse more often in the

high probability (high magnitude

amplitude)

regions.

These

correspond to the bright fringe

regions, which, with enough

individual quanta collapsing on the

screen, are seen by the human eye.

In the many paths approach, for

any particular spot on the screen, we

would add the phases of every

“possible” path from the emission

point, through one slit, to that spot

(

), plus all paths through the

other slit to the spot. See Figure 4.

screen

double slit barrier

various paths

(

source

Figure 4. Double Slit Experiment in

Many Paths Approach

The result would be proportional to the amplitude at the spot found in the Schroedinger

approach. That is, the sum of all phasors at

(see (25)) yields

lim

( , ; , ; )

U x y x y T

→∞

(26)

where

is some constant.

We would then repeat that procedure for every other point on the screen. Since, for a fixed

source at

, and a fixed

for the screen, the amplitude would be spatially only a function of

, and we could express it simply as

(

6.5 Finding the Proportionality Constant: By Example

The square of the absolute value of the amplitude

is the probability density. So we can

normalize

over the length of the screen, i.e.,

lim

( )

U y

=+∞

=−∞

→∞

(27)

and thus, once the value of the limit is determined, readily find the proportionality constant

7 Summary of Approaches

7.1 Feynman’s Postulates

Richard Feynman was probably well aware of much of the foregoing when he speculated on

the viability of the following three postulates for his many paths approach. Subsequent extensive

analysis by Feynman and many others has validated his initial speculation.

The postulates of the many paths approach to quantum theories are:
1.

The phasor value at any final event is equal to

iS/

where the action

is calculated along

a particular path beginning with a particular initial event.

The probability density for the final event is given by the square of the magnitude of a

typically complex amplitude.

That amplitude is found by adding together the phasor values at that final event from all

paths between the initial and final events, including classically impossible paths.

The amplitude of the resultant summation must then be normalized relative to all

other possible final events, and it is this normalized form of the amplitude that is

referred to in 2.

Note two things.

First, there is no weighting of the various path phasors. The nearly classical paths are not

weighted more heavily than the paths that are far from classical. That is, the different individual

paths in the summation do not have different amplitudes (see (24) and Fig. 3). The correlation

with the classical result comes from destructive interference among the paths far from classical,

and constructive interference among the paths close to classical.

Second, time on all paths (all histories) must move forward. This is implicit in the exponent

phase value of (19), where the integral of

is over time, with time moving forward. Our paths

do not include particles zig-zagging backward and forward through time.
Footnote:

Caveat: A famous quote by Freeman Dyson states that Feynman, while speculating on this approach,

told him that one particle travels all paths, including those going backward in time. But the usual development of

the theory (see Section 8) only includes paths forward in time. Perhaps all paths backward in time sum to zero and
so are simply ignored. In such case, Dyson’s quote would be accurate. But I don’t know for sure.

End footnote.

7.2 Comparison of Approaches to QM

Unifying Chart 1 summarizes the major similarities and differences between alternative

approaches to QM.

Unifying Chart 1. Equivalent Approaches to Non-relativistic Quantum Mechanics

Schroedinger Wave

Mechanics

Heisenberg Matrix

Mechanics

Feynman Many Paths

Probability

Density of

Position

Eigenstates

|amplitude|

Transition

Amplitude

( , ; )

iHT

U x x T

x e

−

( , ; )

lim

( )

x f

U x x T

x t

→∞

∝

Comments Above assumes normalized

states.

Same results as

other two

approaches.

RHS above must be
normalized for

We haven’t done the integral

part yet.

8 Finite Sums to Functional Integrals

8.1 Time Slicing: The Concept

After all of the foregoing groundwork, it is time to extend the phasor sum of a finite number

of paths, such as we saw in Fig. 3 and (24), over into an infinite sum, or in other words, an

integral. To do this, we first consider finite “slices” of time, for a finite number of paths in one

spatial dimension, as shown in Fig. 5 where, for convenience, we plot time vertically and space

horizontally. As opposed to our spatially 2D example in Fig. 3, different paths in Fig. 5 actually

refer to the particle traveling the same direction

between

and

, though at varying velocities.

The paths between each slice are straight lines, but there is no loss in generality, as one can take
the time between slices

∆

, and thus, any possible shape path can be included.

As noted earlier, for any single path, the

one path

phasor at

t L

(28)

The amplitude

for the transition from

is proportional to the sum of (28) for all paths,

sum of

phasors at

lim

→∞

∞

(29)

8.2 Space Slicing: Simple Paths with Discrete Approximation

To evaluate (29), we next also discretize (“slice”) space, and consider a small number

(three) of paths over a small number of discrete events in spacetime, as in Figure 6. We label the

paths a, b, and c, and the events with two numbers, such that the first number represents the time

slice, and the second the space slice. The continuous range of

values at time

will be

designated

; at

; etc. We limit the spatial range for paths considered to

–

, where

the number of paths

= 3 =

∆

. Each path passes through the center of one

∆

segment.

We then assume the phase

is zero, and find the phasors at

for each of the three paths

by subsequently adding the phase difference between discrete events along a given path, as in the

second column of Table 2 below. There, as elsewhere herein, clicking on an equation will

enlarge it for easier viewing. Clicking again will contract it to its original size.

Note that in the last line of column two in Table 2, the Lagrangian

without subscript is

assumed to be the

for the particular subpath being integrated, and this is common notation.

In column three, we approximate the integrals of

over

, such that, for example, for path a

over an interval

∆

apprx

≈

∆

(30)

where, for the first subpath,

( )

( ,

)

apprx

V x

x x

−

≈

−

≈

∆

(31)

Similar relations hold for the other subpaths, and are shown in Table 2.

(

)

}

(

)

0,1,2

a b

Figure 5. Time Slicing for Finite

Number of Paths

Figure 6. Space Slicing for Three

Dicrete Paths

Note that (31) is solely a function of

and

02.

The summation of all three paths in the last

row of column three in Table 2 is solely a function of

02,

, and the three intermediate event

values

, and

. Since

and

are the initial and final events, which are fixed and the

same for all paths, the final summation approximation in Table 2 are really only functions of the

three

. It will, however, serve a future purpose if we keep

and

in the relationship for the

time being.

Table 2. Adding Phasors at the Final Event for Three Discrete Paths

Path

Phasor at

in Terms of Approx

(

)

02 11

11 23

dt i

→

(

)

(

)

f x x

−

∆

−

∆

≈

(

)

02 12

12 23

dt i

→

(

)

(

)

f x x

−

∆

−

∆

≈

(

)

02 13

13 23

dt i

→

(

)

(

)

f x x

−

∆

−

∆

≈

Sum

of a,

b, c

dt i

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

02 11

02 12

02 13

02 1

f x x

f x

f x x

The final relationship in Table 2 is approximately proportional to the transition amplitude,

i.e.,

(

)

(

)

(

)

(

)

(

)

02 1

, ;

apprx

f x

f x x

x x

U i f T t

= −

≈

(32)

where

is some constant, and what we designated as a function

in Table 2, in order to

emphasize its independent variables, is actually an approximation to the action

Since

proportional

to the sum of the phasors, we can multiply the RHS of (32) by any

constant we like and the proportionality still holds. To aid us in taking limits to get an integral,
we multiply (32) by

∆

, and get

(

)

(

)

(

)

02 1

, ;

apprx

x x

U i f T

′

≈

∆

(33)

where

is a new constant. Taking the limit where

∆

means taking the number of paths

. And thus,

(

)

(

)

(

)

(

)

(

)

02 1

, ;

lim

apprx

x x

U i f T

dx C

→∞

′

≈

∆

′

≈

(34)

where our discrete values

have become a continuum

, and it is implicit that the

of the last

part of (34) is that over the appropriate path corresponding to each increment of

. (34) is still

only approximately proportional to the amplitude because time is still discretized in

∆

intervals

and we limit the integration range to

. Before extending those limits, however, we

must consider a slightly more complicated set of paths.

8.3 From Simple Discrete Paths to the General Case

In Figure 7 we introduce one more time interval between the initial and final events,

resulting in nine discrete paths.

Repeating the logic from the previous section (use Table 2 as an aide), the phasor of the first

path (02 11 21 33) is simply

(

)

0 2

1 1

2 1

3 3

1 1

2 1

3 3

0 2

1 1

2 1

1 s t

p a t h

o n ly

d t

→

+ +

(35)

We repeat this for the other eight paths, approximate

along subpaths as before, and take

below to indicate the

∆

segment. This results in a phasor summation from all paths at event

(= 33) [compare with last row, last column of Table 2 and (32)] proportional to the amplitude,

i.e.,

(

)

Figure 7. Nine Discrete Paths

between Two Events

(

)

(

)

(

)

(

)

, ;

apprx

x x

U i f T

≈

(36)

Note that (36) depends on the discrete values of both

and

. So, as we did with (33), we can

multiply (36) by one or more constants without changing the proportionality. We choose to
multiply by

∆

and

∆

. We follow by taking limits where

∆

and

∆

(i.e.,

), [compare with (34)] which results in

(

)

(

)

(

)

(

)

(

)

(

)

(

)

02 1

, ;

lim

apprx

x x

U i f T

x x

dx dx

→∞

′

≈

∆ ∆

′

≈

x x

(37)

We can readily generalize (37) to any number of time slices as

(

)

, ;

....

...

Approximation for Transistion Amplitude

x x

U i f T t

dx dx dx

= −

≈

(38)

where, as before, it is implicit that

in the integral is for the particular path that crosses the

respective

slices at

,…

8.4 From Approximate to Exact

To get a precise, not approximate, relation for the RHS of (38) we would have to do two

things.

Take the

range from

to infinity, i.e.,

– and

, and

Take

∆

for the same

(time between events.)

Doing this, (38) would become

(

)

integ limits along with

symbol imply

paths between i and f

, ;

Exact Expression for Transition Amplitude

x x

t L

x x

all

U i f T t

= −

(39)

The symbol , as noted earlier, represents integration over all paths. With this, the integration
limits designate the initial and final

values and do not imply a constraint on the

dimension

during the integration (as was the case with (38).) In (39) we have, at long last, obtained the

relation of integration type #4 in Table 1, where

t L

F e

(40)

8.5 Practicality and Calculations

Practically, for the first approximation addressed in Section 8.4, we really don’t have to take

to infinity, as we know that paths outside of a reasonably large range from the initial and final

spatial locations will sum to very close to zero. So we can live with significant, but not infinite,

For the second approximation, we only need small enough

∆

such that taking a smaller

value does not change our answer much.

If we use (38), with judicious choices for

∆

and

, we can, in many cases, obtain valid

closed form solutions for the amplitude. We can also obtain numerical solutions with a digital

computer by using approximations for

between time slices, as we did previously. That is, we

can approximate the RHS of (38) in the manner we did for the first line of (37), but extending the

approximation of (37) from 2 to

time slices.

9 An Example: Free Particle

We will first determine the amplitude (and thus the detection probability density) of a free

particle via the Schroedinger approach and then compare it to that for Feynman’s many paths

approach.

9.1 Schroedinger Transition Amplitude

Recall, from Section 3.2, that, in the Schroedinger approach, a position eigenstate is a delta

function, and as it evolves, the wave function envelope spreads and the peak diminishes. |

for

such functions is the probability density at the final point

, after time

, where the peak is

located. We should then expect |

to decrease as

increases, and to equal infinity when

= 0.

We start with the Schroedinger transition amplitude relation (9),

( , ; )

iHT

U x x T

x e

−

(41)

Since the bra and ket here are Dirac delta functions, with the well known relation

(

)

(

)

(

)

x x

ik x x

x x

+∞

−

−∞

−

(42)

we can re-write (41) as

(

)

( , ; )

(

)

(

)

iHT

U x x T

x x e

x x dx

∞

−

−∞

−

(43)

(For readers unfamiliar with operators in exponents, one can express the exponential quantity as
a Taylor series expanded about

, i.e.,

(

) =

–

iTH/

= 1 –

iTH/

+ ½

+… Then, operate

on the ket/state term by term [getting terms in

iET

/ to various powers], and finally re-express

the resulting Taylor series as an exponential in

iET

/ . We have taken the ket with time

= 0 to

make things simpler, but even if you like to think of the Hamilton operator as a time derivative,

when it acts on that ket, it functions as an energy operator and still yields the energy.)

For the exponential with the

operator acting on the initial state, and

, (43) is

(

)

(

)

(

)

(

)

/ 2

( , ; )

x x

TH i

x x

iTp

U x x T

dp dx

′

∞

+∞

−

+∞ −

−

−∞

′

∞

+∞

−

+∞

−

−∞

′

(44)

We then re-arrange (44) to get

(

)

/ 2

(

)

/ 2

(

)

( , ; )

p p

p x

iTp

p x

iTp

p p

U x x T

dx e

dp dp

′

−

′

(45)

Using the integral formula

/ 4

ax bx

+∞ − +

−∞

(46)

we find

(

)

( , ; )

m x x

U x x T

−

(47)

The astute reader may question whether (46), with complex

and

, converges. It does because

the integrand oscillation rate increases with larger |

| in such as way as to make successive cycles

shorter. As |

| gets very large, the cycles become so short that the contribution from each cycle

(think area under a sine curve) tends to zero, and it does so in a manner that allows the integral to

converge.

From (47), the probability density at event

( , ; )

U x x T

(48)

which, as we said it must, decreases with increasing

, and equals infinity for

= 0. Note also,

that increasing

increases the envelope height, and thus decreases the width (for constant area

under the envelope = constant probability.) In other words, the wave packet approaches more
classical behavior, i.e., a narrower, more well defined location. Further, if were to go to zero,
the peak would be infinite, i.e., we would have a delta function and an exact particle location, as

in classical mechanics.

9.2 Many Paths Transition Amplitude

We now seek to derive (47) using the many paths approach.
A free, non-relativistic, particle has Lagrangian

(

)

( )

x t

+ ∆ −

≈

∆

(49)

where the RHS is an approximation between adjacent time slices. Taking

= 0, and

, (38)

becomes

(

)

, ;

....

...

....

...

x x

dt i

U i f T

dx dx dx

−

∆

−

=∞

=−∞

=∞

=−∞

≈

(

)

(

)

(

)

(

)

2 ( )

( )

...

x xi

x x

f x

dx dx dx

dx dx

−

∆

−

∆

=∞

=−∞

=∞

=−∞

(50)

where the underbracket notation will help us in subsequent sections.

9.2.1 Background Math

Look, for the moment, at the last two factors (functions

and

) in the integral. They must be

integrated over

, and that result is a function of

. When one of the two functions in such a

procedure is a function of

–

, as it is here, the integral is called a

convolution integral

. (See

http://www-structmed.cimr.cam.ac.uk/Course/Convolution/convolution.html

In mathematical circles (search “Borel’s Theorem”), it is well known that the Fourier (and

also, the Laplace) transform of such an integral equals the product of the Fourier (or Laplace)
transforms of the two functions. That is, for representing Fourier transform,

{

}

{ }

(

) ( )

f x

x f x dx

−

(51)

Note that although

is a function of

, we can write

(

) because

is fixed.

Each factor in the last row of (50), as one moves leftward, plays the part of

in the theorem

above for the next convolution integral, where the prior convolution integral plays the role of

We get, in essence, a series of nested convolution integrals. Using (51), you should be able to

prove to yourself that the transform of (50) equals the product of the transforms of the

exponential factors in (50). If you can’t, or don’t want to bother, proving it, then just accept that

a corollary to (51) is

{

}

{ } { } { }

{ }

....

(

).... (

) (

) ( )

...

f x

x f x

x f x dx dx dx

−

(52)

9.2.2 Evaluating the Amplitude

So, to evaluate (50), using (52), we i) transform each exponential factor

, ii) multiply

those transforms together, and iii) take the inverse transform of the result to get

(actually

U/C

of (50)). This is made simpler, because each

has the same form, so each transform is the same,

i.e.,

{ }

.....

(53)

The Fourier transform of a function

{

}

( )

f x

f p

f x e

∞

−

−∞

(54)

For the

of (50), and for convenience, taking the coordinate

= 0, this is

2( )

( )

i m

f p

∞

−

∆

−∞

(55)

where here and throughout this section,

is merely a dummy variable allowing us to carry out

the math. Using (46), we find (55) becomes

( )

i t

f p

∆

−

∆

(56)

and thus, from (50), (52), and (53),

/ 2

( )

( )..... ( ) ( )

i T

i t

U p

C f p

f p f p

−

∆

≈

(57)

The inverse Fourier transform of (57), is

(

)

/ 2

(

)

( , ; )

( )

p x

i T

p x

U x x T

U p e

i t

∞

−

−∞

−

∞

−

−∞

∆

≈

(58)

In (58), we could have simply used

in the exponent, as we have been taking

= 0, and our

result would have been in terms of

. In that case,

would have been the distance between

and

, i.e.,

–

. In order to frame our final result in the most general terms, we re-introduced

as having any coordinate value in (58).

With (46) again, (58) becomes

/ 2

(

)

( )

( , ; )

i m

i t

U x x T

−

∆

≈

(59)

By comparison with (47), we see the phase and dependence on

is the same as in the wave

mechanics approach. Using that comparison, we can see that the constant of proportionality is

/ 2

( )

i t

∆

(60)

And thus, the probability density at the final event

is the same as (48), i.e.,

( , ; )

U x x T

(61)

where the equal sign is appropriate for

Note that for

v =

(

–

, the amplitude can be expressed in terms of the classical action as

( , ; )

U x x T

(62)

which agrees with (22) if

(

) there equals the root quantity. In the Appendix, we show it does.

9.3 The Message

It has probably not escaped the reader that the evaluation of a free particle using Feynman’s

many paths approach is considerably more complicated and lengthy than the Schroedinger

approach. This is true for most, if not all, problems in QM.

The disadvantages of the many paths approach in QM are these.

It is generally more mathematically cumbersome and time consuming than the wave

mechanics approach.

The quantity calculated is only proportional to the amplitude, and further analysis is

required to determine the precise amplitude.
3.

The approach is suitable primarily for position eigenstates and is not readily amenable to

more general states, so it is generally not as all encompassing in nature.

The advantages of the many paths approach are these.

The approach also applies to quantum field theory (QFT). In a number of instances therein,

development of the theory is more direct, and calculation of amplitudes is easier, than with the

alternative approach (canonical quantization).

Philosophically, we see that there is more than one way to skin a cat. We learn anew that

the physical world can be modeled in different, equivalent ways. We learn caution with regard

to interpreting a given model as an actual picture of reality.

10 Quantum Field Theory via Path Integrals

So far, we have dealt primarily with non-relativistic quantum mechanics (QM), but the many

paths approach is also applicable to relativistic quantum mechanics (RQM), and as noted above,

to quantum field theory (QFT). (RQM is often confused with QFT. For a comparison of the

similarities and differences between the two, see

Quantum Field Theory: A Pedagogic Intro

Further similarities and differences are illustrated in Unifying Chart 2, below.)

We will not go deeply into QFT, and only outline, in a broad overview, how the theory

presented herein is applicable therein. This should help those students who continue on to the

standard texts for the many paths approach keep the forest in view while examining the trees.

10.1 Particle Theory (QM) vs Field Theory (QFT)

For the many paths approach, we want to make the jump from QM, which is a quantized

version of particle theory, to QFT, which is a quantized version of field theory. Unifying Chart 2

below can help us do that. In it, the 2

and 3

columns compare particle theory

entities/concepts to corresponding field theory entities/concepts. The upper half of the chart, as

indicated, summarizes classical theory (non quantum, and implicitly including special relativity).

The lower half summarizes quantum theory

via approaches other than many paths

. The chart

should be relatively self explanatory, so we will not comment much on it.

We compare the quantum approaches of Unifying Chart 2 to the many paths approach in the

next section.

Unifying Chart 2. Comparing Particle Theory to Field Theory:

Classical and Quantum

Particle Theory

Field Theory

Classical Theory

Indep variables 1D 3D

t t

x,y,z,t

Depend

variables

(

)

(

)

position

(

x,y,z,t

)

field

Dynamic

variables

(functionals)

Particle total value:

functions of , ,

(or , , )

x x t

r r

Density values (per unit vol):

, ,

functions of , , , , ,

x y z t

φ φ

, etc.

d x

Equations of

motion

or equivalently, Euler-Lagrange

formulation,

∂

−

∂

(force/vol) for media;

Maxwell’s eqs for e/m,

or equivalently,

for of either,

∂

−

∂

Variable

correspondences

particle field

t x,y,z,t

total values density values

Quantum Theories

QM and RQM via

Wave Mechanics

QFT via Wave Mechanics =

Canonical Quantization

Quantum

character

change

and all dynamic variables

operators

and all dynamic variables

operators

New quantum

entity

state

wave function

state

different from

(operator) field

Note

Fields create & destroy states. States

can be multiparticle (

, ...

φ φ

)

Operators

functions of , ,

x x t

functions of , ,

φ φ

Expectation

values of

operators

etc. for other opers

or for multiparticle state

, ...

φ φ

Equations of

motion

For wave function

QM: Schroedinger eq

RQM:

Klein-Gordon,

Dirac,

Proça eq

or equivalently,

Euler-Lagrange formulations

For quantum field

QFT: Klein-Gordon, Dirac, Proça eqs

or equivalently,

Euler-Lagrange formulations

Macro

equations of

motion

Deduced from above and expectation

values of force, acceleration

Deduced from above and expectation

values of relevant quantities

Transition

amplitude

( , ; )

iHT

U x x T

x e

−

are eigen states of position

( , ; )

iHT

φ φ

−

states can be multiparticle

probability density

probability

10.2 “Derivation” of Many Paths Approach for QFT

From the last row of Unifying Chart 2, we see that the transition amplitude for the QFT

canonical approach, which is essentially a wave mechanics approach for relativistic fields, is

similar in form to that of the QM wave mechanics approach, given that we note the
correspondence

between QM and QFT. An additional fundamental difference between the

two is the form of the Hamiltonian

. In QM,

is a non-relativistic function of

x, dx/dt,

and

(rarely)

. In QFT, it is a relativistic function of

, and (rarely)

Since the canonical (wave mechanics) QFT approach mirrors the wave mechanics QM

approach, one could postulate (and Feynman probably did) that the many paths approach in QFT

would mirror the many paths approach in QM. (See Unifying Chart 1 in Section 7.2 for the

corresponding QM transition amplitudes using each approach.) Simply using the same
correspondences

and

nonrel

rel

(and thus,

nonrel

rel

) for the many paths approach

yields Unifying Chart 3.

Unifying Chart 3. Comparing QM to QFT for the Many Paths Approach

Quantum Theories

QM and RQM via

Many Paths

QFT via

Many Paths

Transition

amplitude

( , ; )

lim

( )

x f

U x x T

x t

→∞

∝

( , ; )

lim

( )

d x

φ φ

→∞

∝

Note

Above is from Unifying Chart 1 in

Section 7.2

Above is a simplified example for a

single scalar field.

In the RH column above, all paths, comprising all configurations of the entire field

over all

space between its initial and final configurations, are added (integrated).

here is the action for

the entire field. is the (relativistic) Lagrangian

density

for the field, which, integrated as it is

above over all space

, yields

Of course, the many paths transition amplitude of Unifying Chart 3 is, at this point, only a

guess. However, decades of research, first by Feynman and then by many others, has proven that

it is completely valid.

To summarize, briefly

Unifying Chart 4. Super Simple Summary

Correspondences

nonrel

rel

Wave mechanics amplitude

QM QFT

canonical quantization QFT

Many paths amplitude

QM QFT

functional quantization QFT

10.3 Time Slicing in QFT

Using the same correspondences as in Unifying Chart 4, and the time slicing approximation

for QM of (38), we find, for QFT,

(

)

, ;

....

...

QFT Approximation for Transition Amplitude

d x

U i f T t

d d

φ φ

= −

≈

(63)

where the subscripts refer to different time slices, not to different fields. This example is for

only a single field.

The exact form of the transition amplitude, obtained from (39), is given in Unifying Chart 3,

and is repeated here,

( , ; )

( )

QFT Exact Expression for Transition Amplitude

d x

U i f T

(64)

10.4 More Ahead in Path Integral QFT

Of course, we have only scratched the surface of the many paths approach to QFT. There is

a great deal more, including some fairly fundamental concepts. However, hopefully, all of the

above will provide a solid foundation for that, by explaining more simply, more completely, and

in smaller steps of development what traditional introductions to the subject often treat rapidly

and in somewhat less than transparent fashion.
––––––––
If you find errors (typographical or otherwise), or have suggestions on how to make anything

herein easier to understand, please help those who come after you by letting me know, so I can

make appropriate corrections/modifications. I can be reached via the email address in the home

page,

Pedagogic Aides to Quantum Field Theory

, for which this material is a sub-section.

Thank you.
Distribution of this material to others is encouraged though subject to (fairly liberal) copyright

restrictions. These can be found at the above link as well.

Robert D. Klauber

April 30, 2009

Appendix

From (20), with

x=x

peak

(

)

(

)

( )

(

, )

( )

peak

i t

p p

Et px

A t

g p e

+∞

−∞

−

(65)

We note that (20), and thus (65), are derived from the general wave packet relation (see ref 2)

(

)

( , )

( )

i Et px

x t

g p e

+∞

−∞

−

(66)

At our initial event, take

t=t

= 0, so the above becomes

( ,

( )

x t

g p e

+∞

−∞

= =

(67)

If (67) is a delta function centered at

= 0, then, from the definition of the delta function,

( ,

( )

i px

x t

+∞

−∞

= =

(68)

Comparing (68) to (67), we see that for an initial delta function measured at

( ) 1

g p

(69)