When people talk about quantum mechanics, they’re usually referring to applying the principles of quantum mechanics to *particles* which are moving with *non-relativistic* energies. That’s fine, and quantum mechanics (or really, non-relativistic quantum particle mechanics) is extremely useful for huge ranges of problems, but there are also whole fields of physics for which we *must* take special relativity into account. Further, we know that particles are created and destroyed all the time—particle accelerators produce showers of daughter particles, which themselves decay into still other particles, and so on. So there is a definite need for some quantum framework which can account for Einstein’s special relativity, as well as for particle creation and destruction. This encompassing quantum framework is known as *Quantum Field Theory*.

In developing a relativistic quantum theory, it turns out that we can’t just quantize relativistic particles the way we can quantize non-relativistic particles, as I’ll detail below. Rather, we will need to construct a quantum *field* in order to take quantum mechanics into the relativistic regime. Here, we’ll take it for granted that a quantum field can be constructed which satisfies certain fundamental requirements.

In this notation, is a 4-vector representing a point in spacetime, is a spatial vector in , and I’ll use boldface for operators (like the Hamiltonian, ). In the end our quantum field () will be operator-valued at each spacetime point—I just won’t use boldface for the fields themselves, even though they are operators. So associates each spacetime point with an operator called .

So, why doesn’t familiar quantum mechanics play well with relativity?

*The easy way out and negative energy*

In non-relativistic quantum mechanics, the Schrödinger equation (SEq),

,

describes the evolution of a particle’s state in time. Unfortunately for physicists, the SEq is not “relativistically invariant.” This means that if a wave function satisfies the SEq in one reference frame, it will not generally satisfy the equation after we perform a Lorentz boost or a rotation to a different reference frame.

There is a wave equation, however, which is relativistically invariant and does naturally arise as the description for a certain type of quantum field (although here we’ll treat it as an equation for a classical field ). This is the Klein-Gordon equation,

,

which looks just like a classical wave equation with an added mass term. We can solve the Klein-Gordon equation, starting with an educated guess for :

Using the Klein-Gordon equation,

Solving for our allowed energies, we see that , and our solution takes the form

( and are just normalization constants).

There’s something fishy going on here: the term corresponds to solutions with *negative *energy. That means treating the Klein-Gordon equation as the equation of motion for a single relativistic quantum particle cannot be the correct interpretation. If is viewed as a field, the negative energy solutions can be interpreted as antiparticle solutions–take a look at Section 8.1 of Sakurai’s “Modern Quantum Mechanics” for more on this interpretation. In any case, a single-particle interpretation of is incorrect.

*Problems come in pairs*

We know from Einstein’s mass-energy equivalence *E* = mc^{2} that mass can be created if a suitable cost in energy is paid. If we scatter two pions, for example, we may expect the following process:

That’s not too exciting on the surface. But if the incoming pions have sufficiently high energy, we will have no choice but to consider the processes

and , and so on,

because the energy of the incoming pions may be high enough to produce numerous daughter pions.

Generally, we have to worry about pair-production of particles whenever a particle is confined to a very small space–comparable to its Compton wavelength , where *m* is the particle’s mass. From the uncertainty relation , if the particle is confined so that the uncertainty in its position is , then . From *E* = mc^{2}, the threshold on momentum for creating a particle is . So if approaches as is squeezed, pair-production is a relevant problem.

Incidentally, this problem of pair-production might make it seem silly to study the hydrogen atom in the scrutinizing detail that non-relativistic quantum mechanics affords it. After all, the electron is confined to an *awfully* small space. But it turns out that the characteristic size of hydrogen’s orbitals is around the Bohr radius, , which can be related to the Compton wavelength of an electron by the fine-structure constant : . So is sufficiently large that pair-production doesn’t come into play, and we can use non-relativistic quantum mechanics to study the hydrogen atom.

Anyway, whatever quantum field theory we use, it must be constructed to account for creation and destruction of particles.

*Faster than light?*

* *We run into another problem in single-particle quantum mechanics when we calculate the amplitude (which represents how probable it is for a process to occur) for a particle to travel from a point to in a time *t*. Because the speed of light *c* is our universal speed limit, we would expect to be zero for points which are separated by a distance farther than light could travel in a time *t*. In special relativity, the boundary of points which are accessible without exceeding *c* defines a *light cone*, shown in Figure 1.

**Figure 1.** For a particle or person at the origin, the speed limit *c* means that only points within the light cone are accessible. It is impossible to travel or send information to points outside the light cone. For you here on Earth, a spot on Jupiter one second in the future is outside your light cone. You cannot influence that spacetime point—no matter what.

So if , the particle cannot possibly travel from to in a time *t*.

*intervals*, as they incorporate both space and time. The spacetime interval between two events is independent of reference frame, so it’s very useful. The interval is defined as (in the sign convention I’m using), where and are respectively the spatial and temporal distances between two events. If two events have a spacetime interval between them, they are said to be

*spacelike-separated*. For spacelike-separated events we have , meaning the events are separated by enough space that one cannot have a causal influence on the other.

Returning to the problem of particles traveling outside their light cone: if we actually use the free-particle Hamiltonian to calculate the amplitude for a particle to travel from to , we start with

.

Inserting an identity operator gives

,

and the exponential term acts on to give

.

The inner-product term is just the wave function for a momentum eigenstate, . After replacing similarly, we have our result:

,

which is nonzero. That means that in the familiar framework of quantum mechanics, it’s possible for a particle to travel between any two points in any amount of time. Even if you replace the Hamiltonian operator in the calculation above with its relativistic form , you still find that is nonzero outside the light cone! That’s no good; faster-than-light travel has to be prohibited by QFT.

As an aside, notice that the we calculated above corresponds to the propagation of a free particle between any two points in space. Strictly speaking, represents the final state of interest and represents the actual final state of the system which starts out localized at , so taking their inner product tells us the overlap of the desired final state with the actual final state. In other words, squaring gives the probability (density) of the propagation from to . Our result for is a 3D Gaussian centered around the initial point, which flattens out as time moves on. That makes some sense; a free particle initially localized at will have a totally undetermined momentum (from the uncertainty principle), and so its motion will not have a preferred direction.

*Long-distance commutation*

* *As a final point in the problem of applying special relativity to quantum mechanics, we’ll take a look at causality from the point of view of *operators*.

First, remember that spacelike-separated points in spacetime cannot have a causal influence on each other.

In non-relativistic quantum mechanics, every Hermitian operator is associated with an observable quantity (say, the component of angular momentum in the z-direction). If two such operators A and B commute, then any observer can measure the quantities associated with A and B simultaneously with well-defined results. This is not so for non-commuting observables such as position and momentum.

Now think about two observers in spacelike-separated regions R_{1} and R_{2}. The experimenter in R_{1} is trying to measure the observable A_{1}, and the exprimenter in R_{2} is trying to measure an the observable A_{2}. If A_{1} and A_{2} commute, the experimenters can succeed.

But what if A_{1} and A_{2} do not commute? Well, then the outcomes of the measurements will depend on the *order* in which the experimenters make their respective measurements—as is necessarily the case with non-commuting observables—meaning that a measurement taking place in R_{1} *superluminally* affects what outcomes may be seen in R_{2}.

_{1z}of particle 1 and S

_{2z}of particle 2, where particles 1 and 2 are in an entangled state). These operators are said to operate in different “subspaces”, and are defined to commute, so both S

_{1z}and S

_{2z}can be simultaneously known. That is, measuring S

_{2z}after measuring S

_{1z}doesn’t change the system’s state

*out*of an eigenstate of S

_{1z}. But if the operators A

_{1}and A

_{2}don’t commute, measuring A

_{1}in R

_{1}and subsequently measuring A

_{2}in R

_{2}will change the system out of a state for which A

_{1}is definite. If an observer in R

_{1}then measures A

_{1}again (after A

_{2}has been measured), she may see a different result than she saw previously, and know superluminally that a measurement was made in R

_{2}which disturbed her system. There is no analog for her obtaining this information in standard entangled measurements. Measuring non-commuting A

_{1}and A

_{2}in spacelike-separated violates causality.

So QFT has a new requirement:

for any observables A_{1} in R_{1} and A_{2} in R_{2}, where R_{1} and R_{2} are spacelike-separated regions. **Spacelike-separated operators must commute.**

This requirement squares nicely with a framework of fields rather than particles. In a field theory, we can treat the fields themselves as operators which depend on spacetime points like and . Any observable quantities we want to measure will be built up of the fields. Then, if we want operators corresponding to spacelike-separated observables to commute, we just have to enforce that the fields themselves commute:

for any spacelike-separated points and .

This is one of about four fundamental requirements for constructing a quantum field. The others deal with their behavior under spacetime translations and Lorentz transformations, as well as the specifics of how they handle creation and annihilation of particles. But that’s a story for another post! Quantum mechanics has totally failed its special relativity exams.

**Recommended Readings:**

[1] Sakurai, JJ; Napolitano, Jim. *Modern Quantum Mechanics*. Chapters 3, 8.

[2] Peskin, Michael E.; Schroeder, Daniel V.. *An Introduction to Quantum Field Theory*. Chapters 1, 2.