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Why quantum field theory?
It is often said that quantum field theory is the natural marriage of Einstein’s special theory of relativity and the quantum theory. The point of this
section will be to motivate this statement. We will, first of all, put forward
an attempt to convince you that there is a conflict between the quantum
mechanical theory of a single particle and the special theory of relativity,
that is, that the two are fundamentally incompatible. Then, we will discuss
how quantum field theory can come to the rescue and resolve this conflict.
The upshot is the statement that, if you want to use quantum mechanics
and you want it to be relativistic, you need a generalization of single-particle
quantum mechanics which contains states with different numbers of particles.
For this, quantum field theory turns out to be an adequate solution to the
problem.
Before we get on with it, though, let us also state that, in the near future,
we will see a construction that shows us that quantum field theory is also
a natural framework in which we can study quantum mechanical systems
where different quantum states can have different numbers of particles. In
fact, this can be very useful even in non-relativistic physics where we are
not forced to use it, or at least there is no basic principle which tells us we
must use it, but where it turns out to give us a convenient approach to many
physical problems.
The idea that a significant generalization of single-particle quantum mechanics might be what is needed in order to describe particles in a way that
is compatible with the special theory of relativity might already be apparent
to the reader. A very important part of special relativity is the equivalence
of energy and mass, as in Einstein’s famous formula E = mc2 . According
to this formula, as far as special relativity is concerned, a particle can be
replaced by some energy. More concretely, we could consider a particle and
an anti-particle. In relativistic physics, they can combine and annihilate,
leaving behind a puff of energy, in the form of photons. To describe this
physical process, we would need a framework wherein there is a quantum
state with a particle and an anti-particle, for example, an electron and a
positron, and another distinct quantum state where there is just the product
of their annihilation, some photons. This is a first criterion. Secondly, we
would also need to be able to analyze the transition between the two states,
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for example, to predict the probability amplitude for this transition to occur.
We will eventually see that a natural framework where both of these criteria
are satisfied is quantum field theory.
The above argument might already be enough motivation to study quantum field theory. However, here, we want to emphasize the point by asking
what happens if we simply generalize what we know about single particle
quantum mechanics to the situation where the particle can have very large
energies, so that its total kinetic energy can have an order of magnitude that
is comparable to its rest mass. As we shall see, we will encounter problems
when we try to describe a single particle which obeys the rules of quantum
mechanics and also obeys the rules of the special theory of relativity.
Let us begin by considering a single particle traveling on open, infinite
three dimensional space. This particle should have a conserved linear momentum, p~, and a conserved total energy, E. According to special relativity,
the energy of the particle and momentum of the particle are related to each
other by the equation
p
(1)
E(~p) = m2 c4 + p~2 c2
where m is the rest mass of the particle and c is the speed of light. In the
quantum mechanics of a single particle, we could consider a quantum state
of the particle which is an eigenstate of its linear momentum,
p̂i |pi = pi |pi , i = 1, 2, 3
(2)
where we have denoted the momentum operator by p̂, with Cartesian components p̂i , the eigenvalue of the momentum operator by pi and the eigenstate
of the momentum operator with eigenvalue pi by the ket |pi. The states with
definite momentum have a continuum normalization, so that the product of
a bra and a ket is
hp|p0 i = δ 3 (~p − p~ 0 ) .
Because the energy of the particle is a simple function of the momentum
momentum of the particle, given in equation (1) above, an eigenstate of the
momentum is also an eigenstate of the energy, that is
p
(3)
H |pi = m2 c4 + p~2 c2 |pi
where H is the Hamiltonian. Here, we are assuming that there is a Hamiltonian operator whose eigenvalues are the energy states of the particle. Once
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we have the Hamiltonian, we can form the Schrödinger equation which must
be satisfied by the time-dependent state vector of the quantum mechanical
particle. We will denote this state by |Ψ(t)i and the Schrödinger equation is
ih̄
∂
|Ψ(t)i = H |Ψ(t)i
∂t
(4)
The solution of this equation, assuming that at t = 0 the particle is in an
eigenstate of momentum p~ is
√
2 4
2 2
Ψ(~p, t) = e−i m c +~p c t/h̄ |pi
(5)
for any value of the momentum vector p~.
This simple development would seem to be a complete solution of the
quantum theory of a single relativistic particle. It allows us to ask any
question about it. For example, let us consider the scenario where, at some
initial time, say t = 0 the particle is localized at ~x = 0. This state could
be created by a measurement of the position of the particle, for example,
which we are assuming can have arbitrarily good resolution. We construct
an eigenstate of position, that is, one which obeys
x̂i |xi = xi |xi
(6)
by superposing the complete set of momentum states as
Z
|~xi = d3 p |~pi h~p | ~xi
(7)
where the overlap matrix is a plane wave
h~p | ~xi =
ei~p·~x/h̄
3
(2πh̄) 2
(8)
Then, the wave function that evolves from an eigenstate of position, at a
time t later, is
√
Z
p2 c2 t/h̄+i~
p·~
x/h̄
−i m2 c4 +~
e
iHt/h̄
3
e
|~xi = d p |~pi
(9)
3
(2πh̄) 2
We can now ask the question as to the amplitude for observing the particle
at point ~y after a time t has elapsed. The answer is simply the overlap of
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the position eigenstate |~y i with the above wave function evaluated at t. The
result is
√
Z
−i m2 c4 +~
p2 c2 t/h̄+i~
p·(~
x−~
y )/h̄
e
iHt/h̄
3
h~y | e
|~xi = d p
(10)
(2πh̄)3
Now, we find the difficulty.1 One of the postulates of the special theory of
relativity states that the speed of light is a maximum speed. However, from
equation (10), the probability amplitude is nonzero in the causally forbidden
region, where |~y −~x| > ct. There seems to be a nonzero amplitude for motion
at speeds greater than that of light.
A formal way to see that (10) is indeed nonzero in the forbidden region,
is to consider t where it occurs in that equation as a complex variable. Then,
(10) is analytic in the lower half of the complex t-plane. When t is real, the
expression is a distribution which should be defined by its limit as complex t
approaches the real axis from the lower half plane. Given that it is analytic
in this domain, it cannot be zero in any region of the lower half plane plus
the real axis except for discrete points, otherwise it would have to be zero
everywhere. It is definitely not zero for all times, in fact when t = 0 it is a
Dirac delta function. Thus, it cannot be zero in the entire region ct < |~y −~x|.
To see this more explicitly, we can do the integral for the special case
where m = 0. It becomes
Z ∞
1
iHt/h̄
pdp e−ip[ct−|~x−~y|]/h̄ − e−ip[ct+|~x−~y|]/h̄
h~y | e
|~xi = 2 2
4π h̄ i|~x − ~y | 0
∂
1
1
1
= lim+ i
−
→0
∂(ct) 4π 2 |~x − ~y | ct − |~x − ~y | − i ct + |~x − ~y | − i
∂
1
P
2
2
=
−iπδ((ct) − |~x − ~y | )sign(t) +
(11)
∂(ct) 2π 2
(ct)2 − |~x − ~y |2
In the second line above, we have defined the integral over the semi-infinite
domain by introducing the positive infinitesimal parameter . This is tantamount to defining the distribution that we obtain as a limit as t approaches
the real axis from the lower half of the complex plane. We have used the
identity
1
P
= + iπδ(x)
x − i
x
1
This is in addition to the already obvious difficulty that the expression (10) is not
Lorentz invariant. In fact, it transforms like the time derivative of a Lorentz invariant
function. Let us overlook this issue for the time being.
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where P/x is the principal value distribution. Also, for the Dirac delta
function
1
δ(x2 − a2 ) =
(δ(x − a) + δ(x + a))
2|a|
In equation (11), we see that the wave-function of a massless particle spreads
in two ways. The first is a wave which travels at the speed of light and is
therefore confined to the light cone - where |~x − ~y | = ct. The second is a
principle value distribution which is non-zero everywhere, including in the
forbidden region where |~x − ~y | > ct. This latter spreading of the wave packet
violates causality. It tells us that, in our quantum mechanical system, the
result of a measurement of the position of the particle at position ~y after time
t could have a positive result. The particle could be observed as travelling
faster than light. This would certainly seem to be incompatible with the
principles of the special theory of relativity where objects are restricted to
having subluminal speeds.
Now that we have found a difficulty with causality, we need to find a way
to resolve it. We will resolve it by going beyond single-particle quantum mechanics to an extended theory where there is another process which competes
with the one that we have described. The total amplitude will then be the
sum of the amplitudes for the two processes and we will rely on destructive
interference of the amplitudes to solve our problem, that is, to make the
probability of detecting the particle identically zero in the entire forbidden
region |~x − ~y | > ct.
To include the second process, we will first frame the first process, the one
we have discussed so far, as the following thought experiment. One observer,
whom we shall all Alice, is located at position ~x and prepares the particle in
the state which is localized at ~x. Alice could do this by measuring the position
of the particle and we assume that the result of the measurement is that the
particle is at position ~x. We assume that Alice can do this measurement with
arbitrarily good precision. Immediately after the measurement, the particle
is allowed to evolve by its natural time evolution, the one which we have
described above, so that after time t, its quantum state is given by equation
(9) and its wave-function by equation (10). Then, at time t, another observer,
Bob, who is located at point ~y does an experiment to detect the particle. Of
course, in a given experiment, Bob might or might not find the particle at
~y . But, given that the particle is has non-zero amplitude to propagate there,
if Alice and Bob repeat this experiment sufficiently many times, Bob will
eventually detect the particle at ~y . The result of the experiment is to collapse
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Figure 1: The wave packet is initially localized at ~x and as time evolves it spreads
in such a way that there is a nonzero amplitude for detecting it in the vicinity of
point ~y . If it is detected at ~y , since |ct| < |~x − ~y |, its classical velocity would be
greater than that of light.
the particle’s wave function to one which is localized at ~y . The amplitude
for the particle to propagate to ~y is given by (10). If this were all there is to
it, the result of the experiment violates causality.
The second process that we will superpose with the one that we have
described will require other states to be introduced. It then clearly involves
an extension of single particle quantum mechanics. In the second process, the
attempt by Bob, the observer who is located at ~y , to observe an electron’s
position creates a pair consisting of a particle and an anti-particle. The
position measure collapses the wave function of the particle into the position
eigenstate localized at ~y , the position which was the final state of the particle
in the first experiment. The anti-particle is interpreted as a particle which
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Figure 2: We should add to the amplitude for the particle to travel from ~x to ~y
as in figure 1 the amplitude that a particle-anti-particle pair is created at ~y , the
particle continues forward in time as it did in the first process, the anti-particle
propagates backward in time and annihilates the particle which was prepared in
the state localized at ~x.
moves backward in time, from time t to time 0.2 After time −t it has an
amplitude to arrive at position ~x where it annihilates the particle that Alice,
the observer at ~x, has prepared in the localized state. The result of this
second process is the same as that of the first process, a particle begins in a
state localized at ~x and after a time t it is detected in a state localized at ~y .
The amplitude for the second process is similar but not identical to that of
the first process, due to the fact that the positron propagates backward in
2
The interpretation of the anti-particle as a particle which moves backward in time
originates with Stueckelberg [1] and it was adopted by Feynman [2] in his formulation of
quantum electrodynamics.
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time. It is
√
−iHt/h̄
h~x| e
i
Z
|~y iantiparticle =
3
dp
e
m2 c4 +~
p2 c2 t/h̄−i~
p·(~
x−~
y )/h̄
(2πh̄)3
(12)
The total amplitude is the sum of amplitudes of the two processes,
A = h~y | e−iHt/h̄ |~xiparticle + h~y | eiHt/h̄ |~xiantiparticle
Z
i
√
ei~p·(~x−~y)/h̄ h i√m2 c4 +~p2 c2 t/h̄
−i m2 c4 +~
p2 c2 t/h̄
= d3 p
e
+
e
(2πh̄)3
(13)
(14)
Now, the total expression can have destructive interference. We will not
demonstrate it in the general case, but in the case where the mass of the
particle and antiparticle is zero. There, we can perform integral in (12)
explicitly,
1
P
∂
2
2
−iHt/h̄
−iπδ((ct) − |~x − ~y | )sign(t) −
h~x| e
|~y iantiparticle =
∂(ct) 2π 2
(ct)2 − |~x − ~y |2
(15)
We see that, like the amplitude for the particle, the amplitude for the antiparticle also spreads outside of its light cone. However, when we add the
amplitudes of the two processes together, their sum is
h~y | e−iHt/h̄ |~xiparticle + h~x| eiHt/h̄ |~y iantiparticle =
∂ 1 2
2
−iπδ((ct)
−
|~
x
−
~
y
|
)sign(t)
∂(ct) π 2
(16)
We see that the principal value part of the expression, which was nonzero
outside of the light cone, has canceled. What remains describes the wave
function of the initial particle spreading along its light cone, as we might
expect for a massless particle, which travels at the speed of light. This has
the further interesting consequence. The amplitude for the massless particle
to travel at any speed other than that of light vanishes.
The upshot of the above development is that a correct treatment of a
quantum mechanical particle which also obeys the laws of special relativity
requires more than just single particle quantum mechanics. The resolution
of the difficulty that we have suggested needs an anti-particle. Quantum
field theory will supply us with an anti-particle. Another lesson is that
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the properties of the anti-particle must be finely tuned to be very similar
to that of the particle. Otherwise the exact cancellation of the amplitude
outside of the light cone would not happen. We will eventually see that this
fine-tuning is generally a property of the relativistic wave equations which
replace the Schrödingier equation. They have both positive and negative
energy solutions which we shall interpret as belonging to the particle and the
anti-particle that the wave equation simultaneously describes. We will put
off further discussion of this fact until we study wave equations.
References
[1] E.C.C. Stueckelberg, Helvetica Physica Acta, 14, 51 (1941);
E.C.C. Stueckelberg, Helvetica Physica Acta, 15, 23 (1942).
[2] R.P. Feynman, Phys. Rev. 74, 939 (1948).
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