Least Action

Nontrivializing triviality..and vice versa.

Archive for June 2009

Klein Gordon FieldĀ PartĀ II

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In the last post, we looked at how a general solution to the sourced KG equation can be written in terms of a source-free KG solution and the Green’s function of the KG field. The important thing to note here is that the Green’s function depends only on the structure of the equation, specifically, on the differential operator (\Box + m^2), and not on the source J(x). Note also, that x is short for x_{\mu}, the position 4 vector. In our notation \bf{x} will denote the position 3 vector.

The first step in our analysis is to solve for the propagator. To do, we write G(x-x') in terms of its Fourier transform G(p):

G(x-x') = \int \frac{d^{4}p}{(2\pi)^4} e^{-ip\cdot(x-x')} G(p)

where the different components of p^{\mu} should be treated as independent, unrelated by the energy momentum relation. Right now, we’re only at the level of mathematics, so we do not enforce p^{\mu}p_{\mu} - m^2 = 0. Let us operate on the above equation by (\Box_{x} + m^2). This gives us

(\Box_{x} + m^2) G(x-x') = \int \frac{d^{4}p}{(2\pi)^4} e^{-ip\cdot(x-x')}(-p^2 + m^2) G(p)

Using the integral representation of the Dirac delta distribution, we get

G(p) = \frac{1}{p^2-m^2} = \frac{1}{(p^{0})^2-|{\bf{p}}|^2-m^2} = \frac{1}{(p^{0})^2-E_{p}}

where we have used the energy momentum relation in the last two equalities. Putting this back into the analysis equation, we get

G(x-x') = \int \frac{d^{4}p}{(2\pi)^4} e^{-ip\cdot(x-x')}G(p)

Do you see a problem with this expression?

The integration over p^0 cannot be performed as the integrand has poles at p^{0} = \pm E_{p}. This is where Feynman’s genius comes in…we’ll take a look at it in the next blog post!

Written by Vivek

June 22, 2009 at 11:13

Klein Gordon Field Part I

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I’m trying to gain a deeper insight into the Green’s function for the Klein Gordon field, which is used to define the propagator. Peskin and Schroeder (P&S) have provided partial motivational details, and I am trying to complete the argument. Posts that will follow in a similar fashion will be primarily for new students of quantum field theory, like myself, and not for experts or students who have already worked with QFT. So if you belong to the latter category, you should find nothing new here.

For newcomers to quantum field theory, the Klein Gordon Field is a solution to Klein Gordon’s equation:

(\Box + m^2)\phi = J(x)

where as usual, \Box is the d’Alembertian operator and m is the mass of the scalar field (for now, if we consider a real Klein Gordon field, this is just the pass of the particle excitation mode of the field). Note that the “free” Klein Gordon field is one for which the source term J(x) is zero. A general solution to the above equation can be written in terms of the Green’s function for the field, defined by

(\Box_{x} + m^2)G(x-x') = -\delta^{4}(x-x')

where \Box_{x} operates only on terms involving x (and not on x'), and the right hand side is our usual Minkowski space Dirac delta distribution. As stated, the Green’s function provides a concise way of writing a general solution to Klein Gordon equation with a source term:

\phi(x) = \phi_{0}(x) - \int d^{4}x' G(x-x')J(x')

where \phi_{0}(x) is any solution to the free Klein Gordon equation, and can be chosen appropriately to satisfy boundary conditions. The interesting feature of the Klein Gordon formalism in QFT is that it provides a window to several tools and viewpoints that are thereafter used freely, for instance the concepts of particle/antiparticle as a natural outcome of second quantization, Feynman diagrams for scattering amplitudes, propagators isomorphic to Green’s functions and the notion of causality. The free Klein Gordon equation of course does not include interactions and is admittedly “simple”, but as is the case in almost all of science, simple toy models are quite powerful in conveying the features and limitations of a theory, pushing it almost to the fringe. In this framework, we can even gain a glimpse into some of the many infinities plaguing field theory.

In the next blog post, we will look at how the Green’s function nicely comes out of the above formalism with a minimal amount of mathematics. Then, we will discuss a small aspect of Feynman’s brilliance in taking it forward to connect it to the physics of the Klein Gordon field. Have fun!

Written by Vivek

June 22, 2009 at 10:06