Monthly Archives: May 2008

Planck Energy and Perturbation Theory

The known elementary particles range in mass from about 0.0004eV for the lightest neutrino to around 170 billion eV = 1.7 x 10^11 for the heaviest quark, the top. This is a ratio of about 400,000,000,000,000 to 1.

On the other hand, the energy scale available from Einstein’s theory of gravitation (which relates mass to energy) suggests that the natural mass for a typical particle should be the Planck mass, about 2.43 × 10^27 eV, 11 orders of magnitude larger than even the top quark and 25 orders of magnitude larger than the lightest neutrino mass.

From the point of view of the Planck energy, all particles known to man have mass very close to zero. Let’s write the Hamiltonian for the system as a first order Hamiltonian H^{(0)}  , in which the energies (and therefore the masses) of all our usual particles are zero, plus a perturbation \lambda H^{(1)}  , which will provide a correction to the zero energies. For the full Hamiltonian we have:
H = H^{(0)} + \lambda H^{(1)},
where \lambda is a small number.
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Mandelstam Variables and Veneziano Amplitudes

Quantum mechanics is a probabilistic theory. The probabilities are the squared magnitudes of (probability) amplitudes. Amplitudes are computed through perturbation theory, or whatever method one can find. The state of the art method is to use Feynman diagrams to find the amplitudes.

When one considers interactions between two particles, there are two Feynman diagrams that are of particular interest, the “t-channel” and the “s-channel”. The “s” and “t” are Mandelstam variables. These variables define the interaction in a way that automatically preserves Lorentz invariance.

Special Relativity Momenta

This is a relativistic problem, so all momenta are 4-momenta, that is, they are vectors with four components. The time or zero portion of the vector gives the negative energy of the particle, while the space or 1-3 portion of the vector gives its momentum. A 2-particle interaction involves 4 momenta, two for the incoming particles, and two more for the outgoing particles. That’s a total of 4×4 = 16 variables.

Let p_1, p_2, p_3, p_4 be the four momenta. The first two are for the incoming particles, that is, the ones before the interaction, the 3rd and 4th are for the outgoing particles:
Definition of momenta for four particles
Now each of these p_i are a 4-vector. I will write them as p_i^\mu where p_i^0 is the energy of the particle, p_i^1 is the momentum in the x-direction, etc.
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Review: ‘t Hooft’s The Conceptual Basis of Quantum Field Theory

I should be preparing for my 10 minute lecture on classification of hadrons at the American Physical Society’s Northwest meeting on Saturday, but instead I am reading Gerard ‘t Hooft’s beautiful introduction to quantum field theory, The Conceptual Basis of Quantum Field Theory. I like the approach of this book because it concentrates on what I think of as the heart of quantum field theory, Feynman diagrams.

The author begins with classical field theory, but unlike most he defines the solutions to classical field theory with Feynman diagrams! The difference with the quantum theory is that there are no loops; he calls the loops the “quantum corrections” to the classical theory. So all the diagrams are tree diagrams looking something like this:
Tree type Feynman diagram
In the above, the points each correspond to a single point in spacetime. These points are labeled x_j . The G_{49} in the above is a propagator, the Green’s function. Each of the legs carries one of these. The g_{jkl} and \lambda{jklm} are included for the 3-point and 4-point vertices. They are coupling constants. Uh, I’ve been sloppy in the above, please refer to the ‘t Hooft paper for details.
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