Having gone through the fundamental formalism in Lagrangian Mechanics and understanding the Notations, we can now move on to one of the most important subjects in Physics: Classical Field Theory. Since Einstein, it has been witnessed that Field Theory is a powerful way to a unified theory. Although Einstein was never comfortable with Quantum Mechanics, it turns out that using Field theory along with quantisation provides the way to one of the most accurate theories know to mankind: Quantum Field Theory. So let's start!
In what follows, we will use the scale for which ℏ = c = 1. This normalisation is for notational convenience, and these constants can be put back in the relevant equation by dimensional analysis.
From what we know till now, we can write the Lagrangian for a function φ(x) that depends on the space-time co-ordinates:
φ does not vary under Lorentz transformation, as it is not a vector, or in other words, it is a "scalar". Mathematically, we write this as:
Here, φ' is supposed to be the Lorentz transformed version of φ. Furthermore, there is no "spin" attached to this field and hence, we have a scalar spin-0 particle defined with the above equation of motion. But our job is not done yet. In order to actually find the equation of motion, we need an appropriate Lagrangian for this field, at least for the free case, which corresponds to the free particle. As it turns out, this Lagrangian is as defined in Question 3, part (i) of the note "Lagrangian Mechanics: Questions":
First of all, let us study this Lagrangian and try to understand how it is of the right form. First, note the invariant (∂μφ)(∂μφ) term. It represents the differential coefficient of φ w.r.t. all the four space-time components. Hence, it is the "kinetic term" similar to the kinetic term (dx/dt)2 we generally observe in Classical Physics (of course, along with appropriate mass term, etc.). Also note, that unlike in the non-relativistic case, where we only had the derivative of time, here we have the derivative w.r.t. all the four components of space-time. This reflects the "unification" of time and space, as is advocated by Relativity. Now consider the mass term. Compare this to what you would see in the potential term (also recall, that Potentials in Physics represent interactions between particles, like electric potential, magnetic potential, Higgs potential, etc.) of non-relativistic Lagrangians (and hence, the appropriate minus sign). There is another deep relation as to why we attempt to interpret this as the "potential term": as it turns out, it is the Higgs field that are responsible for mass of particles, and when we come to the Lagrangians with Higgs interactions, we will see terms of exactly this form, and hence find some expressions for what the mass of the particle is. For now, whatever that expression is, we have represented as m for brevity.
If you have worked out the problems in the previous note, then you'll know that applying the equation of motion for this particular Lagrangian, we get:
which is the famous Klein-Gordon equation. This was the first relativistic equation derived in field theory, and was supposed to be the equation to describe elementary particles like electrons, protons, etc. However, as it turns out, it does not satisfy some important requirements of fundamental Physics, as we will come to know. This equation is the 4D version of the famous Helmholtz equation in 3D space:
which has the solution
where x is the 3-vector with Cartesian components (x1, x2, x3) and k is the appropriate vector, as appearing in the Helmholtz equation, and A and B are arbitrary constant. What we see here, is that we have two solutions (as we should, for a second order PDE). Drawing from this result, we can easily see that the corresponding solution to the Klein-Gordon equation is:
where, it should now be noted that k and x are 4-vectors, and the 4-vector dot product is now used which is appropriately shown in the second equality in the exponents. Now, the argument of the exponential e represents a travelling wave, with the first term corresponding to a field with positive energy and the second term is the field with negative energy. This implies, that the Klein-Gordon equation has both positive and negative energy solutions, which is not possible, as energy for a free particle is always positive. This is the first problem with this solution. To understand further contradictions, we need to develop the idea of a "conserved charges" of a system. This is itself an extensive topic and we will need a very important theorem of classical field theory, the Noether's theorem, to understand this idea in its full detail.
In a future note, we will consider exactly these ideas, of charges and their associated invariance in the Lagrangian. Some important results will be on deriving the charges for space-time translations, and other charges due to internal symmetries. We might also consider a little bit of formal Group Theory to understand better the idea of internal symmetries.
By Jivesh Kaushal (Admin at PDEP)
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