Some Basics of Relativistic Quantum Field Theory

So far, on this blog, we have introduced the two great pillars of modern physics, relativity (see From Pythagoras to Einstein) and quantum mechanics (see Some Basics of Quantum Mechanics and More Quantum Mechanics: Wavefunctions and Operators). Although a complete unification between these two pillars is yet to be achieved, there already exists such a unified theory in the special case when gravity is weak, i.e. spacetime is flat. This unification of relativity (in this case special relativity) and quantum mechanics is called relativistic quantum field theory, and we discuss the basic concepts of it in this post.

In From Pythagoras to Einstein, we introduced the formula at the heart of Einstein’s theory of relativity. It is very important to modern physics and is worth writing here again:

\displaystyle -(c\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2=(\Delta s)^2

This holds only for flat spacetime, however, even in general relativity, where spacetime may be curved, a “local” version still holds:

\displaystyle -(cdt)^2+(dx)^2+(dy)^2+(dz)^2=(ds)^2

The notation comes from calculus (see An Intuitive Introduction to Calculus), and means that this equation holds when the quantities involved are very small.

In this post, however, we shall consider a flat spacetime only. Aside from being “locally” true, as far as we know, in regions where the gravity is not very strong (like on our planet), spacetime is pretty much actually flat.

We recall how we obtained the important equation above; we made an analogy with the distance between two objects in 3D space, and noted how this distance does not change with translation and rotation; if we are using different coordinate systems, we may disagree about the coordinates of the two objects, but even then we will always agree on the distance between them. This distance is therefore “invariant”. But we live not only in a 3D space but in a 4D spacetime, and instead of an invariant distance we have an invariant spacetime interval.

But even in nonrelativistic mechanics, the distance is not the only “invariant”. We have the concept of velocity of an object. Again, if we are positioned and oriented differently in space, we may disagree about the velocity of the object, for me it may be going to the right, and forward away from me; for you it may in front of you and going straight towards you. However, we will always agree about the magnitude of this velocity, also called its speed.

The quantity we call the momentum is related to the velocity of the object; in fact for simple cases it is simply the mass of the object multiplied by the velocity. Once again, two observers may disagree about the momentum, since it involves direction; however they will always agree about the magnitude of the momentum. This magnitude is therefore also invariant.

The velocity, and by extension the momentum, has three components, one for each dimension of space. We write them as v_{x}, v_{y}, and v_{z} for the velocity and p_{x}, p_{y}, and p_{z} for the momentum.

What we want now is a 4D version of the momentum. Three of its components will be the components we already know of, p_{x}, p_{y}, and p_{z}. So we just need its “time” component, and the “magnitude” of this momentum is going to be an invariant.

It turns out that the equation we are looking for is the following (note the similarity of its form to the equation for the spacetime interval):

\displaystyle -\frac{E^{2}}{c^{2}}+p_{x}^{2}+p_{y}^{2}+p_{z}^{2}=-m^{2}c^{2}

The quantity m is the invariant we are looking for (The factors of c are just constants anyway), and it is called the “rest mass” of the object. As an effect of the unity of spacetime, the mass of an object as seen by an observer actually changes depending on its motion with respect to the observer; however, by definition, the rest mass is the mass of an object as seen by the observer when it is not moving with respect to the observer, therefore, it is an invariant. The quantity E stands for the energy.

Also, when the object is not moving with respect to us, we see no momentum in the x, y, or z direction, and the equation becomes E=mc^{2}, which is the very famous mass-energy equivalence which was published by Albert Einstein during his “miracle year” in 1905.

We now move on to quantum mechanics. In quantum mechanics our observables, such as the position, momentum, and energy, correspond to self-adjoint operators (see More Quantum Mechanics: Wavefunctions and Operators), whose eigenvalues are the values that we obtain when we perform a measurement of the observable corresponding to the operator.

The “momentum operator” (to avoid confusion between ordinary quantities and operators, we will introduce here the “hat” symbol on our operators) corresponding to the x component of the momentum is given by

\displaystyle \hat{p_{x}}=-i\hbar\frac{\partial}{\partial x}

The eigenvalue equation means that when we measure the x component of the momentum of a quantum system in the state represented by the wave function \psi(x,y,z,t), which is an eigenvector of the momentum operator, then then the measurement will yield the value p_{x}, where p_{x} is the eigenvalue correponding to \psi(x,y,z,t) (see Eigenvalues and Eigenvectors), i.e.

\displaystyle -i\hbar\frac{\partial \psi(x,y,z,t)}{\partial x}=p_{x}\psi(x,y,z,t)

Analogues exist of course for the y and z components of the momentum.

Meanwhile, we also have an energy operator given by

\displaystyle \hat{E}=i\hbar\frac{\partial}{\partial t}

To obtain a quantum version of the important equation above relating the energy, momentum, and the mass, we need to replace the relevant quantities by the corresponding operators acting on the wave function. Therefore, from

\displaystyle -\frac{E^{2}}{c^{2}}+p_{x}^{2}+p_{y}^{2}+p_{z}^{2}=-m^{2}c^{2}

we obtain an equation in terms of operators

\displaystyle -\frac{\hat{E}^{2}}{c^{2}}+\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}=-m^{2}c^{2}

or explicitly, with the wavefunction,

\displaystyle \frac{\hbar^{2}}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}-\hbar^{2}\frac{\partial^{2}\psi}{\partial x^{2}}-\hbar^{2}\frac{\partial^{2}\psi}{\partial y^{2}}-\hbar^{2}\frac{\partial^{2}\psi}{\partial z^{2}}=-m^{2}c^{2}\psi.

This equation is called the Klein-Gordon equation.

The Klein-Gordon equation is a second-order differential equation. It can be “factored” in order to obtain two first-order differential equations, both of which are called the Dirac equation.

We elaborate more on what we mean by “factoring”. Suppose we have a quantity which can be written as a^{2}-b^{2}. From basic high school algebra, we know that we can “factor” it as (a+b)(a-b). Now suppose we have p_{x}=p_{y}=p_{z}=0. We can then write the Klein-Gordon equation as

\frac{E^{2}}{c^{2}}-m^{2}c^{2}=0

which factors into

(\frac{E}{c}-mc)(\frac{E}{c}+mc)=0

or

\frac{E}{c}-mc=0

\frac{E}{c}+mc=0

These are the kinds of equations that we want. However, the case where the momentum is nonzero complicates things. The solution of the physicist Paul Dirac was to introduce matrices (see Matrices) as coefficients. These matrices (there are four of them) are 4\times 4 matrices with complex coefficients, and are explicitly written down as follows:

\displaystyle \gamma^{0}=\left(\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right)

\displaystyle \gamma^{1}=\left(\begin{array}{cccc}0&0&0&1\\ 0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{array}\right)

\displaystyle \gamma^{2}=\left(\begin{array}{cccc}0&0&0&-i\\ 0&0&i&0\\0&i&0&0\\-i&0&0&0\end{array}\right)

\displaystyle \gamma^{3}=\left(\begin{array}{cccc}0&0&1&0\\ 0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{array}\right).

Using the laws of matrix multiplication, one can verify the following properties of these matrices (usually called gamma matrices):

(\gamma^{0})^{2}=1

(\gamma^{1})^{2}=(\gamma^{2})^{2}=(\gamma^{3})^{2}=-1

\gamma^{\mu}\gamma^{\nu}=-\gamma^{\mu}\gamma^{\nu} for \mu\neq\nu.

With the help of these properties, we can now factor the Klein-Gordon equation as follows:

\displaystyle \frac{\hat{E}^{2}}{c^{2}}-\hat{p}_{x}^{2}-\hat{p}_{y}^{2}-\hat{p}_{z}^{2}-m^{2}c^{2}=0

\displaystyle (\gamma^{0}\frac{\hat{E}}{c}-\gamma^{1}\hat{p}_{x}-\gamma^{2}\hat{p}_{y}-\gamma^{3}\hat{p}_{z}+mc)(\gamma^{0}\frac{\hat{E}}{c}-\gamma^{1}\hat{p}_{x}-\gamma^{2}\hat{p}_{y}-\gamma^{3}\hat{p}_{z}-mc)=0

\displaystyle \gamma^{0}\frac{\hat{E}}{c}-\gamma^{1}\hat{p}_{x}-\gamma^{2}\hat{p}_{y}-\gamma^{3}\hat{p}_{z}+mc=0

\displaystyle \gamma^{0}\frac{\hat{E}}{c}-\gamma^{1}\hat{p}_{x}-\gamma^{2}\hat{p}_{y}-\gamma^{3}\hat{p}_{z}-mc=0

Both of the last two equations are known as the Dirac equation, although for purposes of convention, we usually use the last one. Writing the operators and the wave function explicitly, this is

\displaystyle i\hbar\gamma^{0}\frac{\partial\psi}{c\partial t}+i\hbar\gamma^{1}\frac{\partial\psi}{\partial x}+i\hbar\gamma^{2}\frac{\partial\psi}{\partial y}+i\hbar\gamma^{3}\frac{\partial\psi}{\partial z}-mc\psi=0

We now have the Klein-Gordon equation and the Dirac equation, both of which are important in relativistic quantum field theory. In particular, the Klein-Gordon equation is used for “scalar” fields while the Dirac equation is used for “spinor” fields. This is related to how they “transform” under rotations (which, in relativity, includes “boosts” – rotations that involve both space and time). A detailed discussion of these concepts will be left to the references for now and will perhaps be tackled in future posts.

We will, however, mention one more important (and interesting) phenomenon in relativistic quantum mechanics. The equation E=mc^{2} allows for the “creation” of particle-antiparticle pairs out of seemingly nothing! Even when there seems to be “not enough energy”, there exists an “energy-time uncertainty principle”, which allows such particle-antiparticle pairs to exist, even for only a very short time. This phenomenon of “creation” (and the related phenomenon of “annihilation”) means we cannot take the number of particles in our system to be fixed.

With this, we need to modify our language to be able to describe a system with varying numbers of particles. We will still use the language of linear algebra, but we will define our “states” differently. In earlier posts in the blog, where we only dealt with a single particle, the “state” of the particle simply gave us information about the position. In the relativistic case (and in other cases where there are varying numbers of particles – for instance, when the system “gain” or “loses” particles from the environment), the number (and kind) of particles need to be taken into account.

We will do this as follows. We first define a state with no particles, which we shall call the “vacuum”. We write it as |0\rangle. Recall that an operator is a function from state vectors to state vectors, hence, an operator acting on a state is another state. We now define a new kind of operator, called the “field” operator \psi, such that the state with a single particle of a certain type, which would have been given by  the wave function \psi in the old language, is now described by the state vector \psi|0\rangle.

Important note: The symbol \psi no longer refers to a state vector, but an operator! The state vector is \psi|0\rangle.

The Klein-Gordon and the Dirac equations still hold of course (otherwise we wouldn’t even have bothered to write them here). It is just important to take note that the symbol \psi now refers to an operator and not a state vector. We might as well write it as \hat{\psi}, but this usually not done in the literature since we will not use \psi for anything else other than to refer to the field operator. Further, if we have a state with several particles, we can write \psi\phi...\theta|0\rangle. This new language is called second quantization, which does not mean “quantize for a second time”, but rather a second version of quantization, since the first version did not have the means to deal with varying numbers of particles.

We have barely scratched the surface of relativistic quantum field theory in this post. Even though much has been made about the quest to unify quantum mechanics and general relativity, there is so much that also needs to be studied in relativistic quantum field theory, and still many questions that need to be answered. Still, relativistic quantum field theory has had many impressive successes – one striking example is the theoretical formulation of the so-called Higgs mechanism, and its experimental verification almost half a century later. The success of relativistic quantum field theory also gives us a guide on how to formulate new theories of physics in the same way that F=ma guided the development of the very theories that eventually replaced it.

The reader is encouraged to supplement what little exposition has been provided in this post by reading the references. The books are listed in increasing order of sophistication, so it is perhaps best to read them in that order too, although The Road to Reality: A Complete Guide to the Laws of Reality by Roger Penrose is a high-level popular exposition and not a textbook, so it is perhaps best read in tandem with Introduction to Elementary Particles by David J. Griffiths, which is a textbook, although it does have special relativity and basic quantum mechanics as prerequisites. One may check the references listed in the blog posts discussing these respective subjects.

References:

Quantum Field Theory on Wikipedia

Klein-Gordon Equation on Wikipedia

Dirac Equation on Wikipedia

Second Quantization on Wikipedia

Featured Image Produced by CERN

The Road to Reality: A Complete Guide to the Laws of Reality by Roger Penrose

Introduction to Elementary Particles by David J. Griffiths

Quantum Field Theory by Fritz Mandl and Graham Shaw

Introduction to Quantum Field Theory by Michael Peskin and Daniel V. Schroeder

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3 thoughts on “Some Basics of Relativistic Quantum Field Theory

  1. Pingback: Some Basics of (Quantum) Electrodynamics | Theories and Theorems

  2. Pingback: Some Basics of Statistical Mechanics | Theories and Theorems

  3. Pingback: An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry | Theories and Theorems

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