This is going to be a bit more haphazard of a blog post, because it was in response to a question I was asked about how String Theory relates to the “universal constants” of physics. I think the best way to describe it is the following sentence:
String Theory provides a single coupling constant that is the expectation value of a scalar field. This constant ends up being independent of the energy scale chosen because this constant (physically) represents the “tension” in the String.
I tried to answer this question without too much math and all of the physics I know (which is pretty much undergrad physics with some knowledge of graduate quantum mechanics, field theory and string theory). The question I was asked was,
I was wondering if string theory had implications towards the values of the ‘universal constants’ and if there was any really obscene math that governed how those values changed over time. When I was first learning physics in HS, I always thought that it was super naive and arrogant even to assume that the universal constant are actually time independent, but maybe the math on universal expansion makes sense with the constants being constants, but I was under the impression string theory offered geometry of hyper dimensions as a reason for these values, possibly the size of curled dimensions as an energy source, etc.
So I replied with something along the lines of the following:
The main result that String Theory provides is that everything boils down to precisely one universal constant. The reason that this is important is because all of the quantum field theories that we know about are effective field theories. What this means is that the specifiers for a field theory, the fields, the equation of motion or an action and the proportionality constants, all depend on the energy regime one is looking at. In the case of the strong nuclear force (which keeps quarks bound to each other, giving rise to mesons such as protons and neutrons), it turns out that at really high energies, the field theory we currently use (Quantum Chromodynamics, QCD) has a coupling constant (called the Yukawa Coupling) that scales roughly as , where is the energy scale (i.e. 1 MeV, 10 GeV, 1 TeV). Note that I’m being a bit sloppy here: The actual scaling is represented by what is known as the function, which is a function of energy scale. However, the idea is that these contributions are only valid within certain energy regimes (i.e. 1 MeV — 10 TeV). In terms of QCD, one of the biggest problems is elucidating how the strong force works at low energies. The current theory gives a lot of infinite integrals that aren’t renormalizable (the heuristic method that Feynman, et. al came up with removing infinites from expectations of electric or magnetic fields) when the energy is low and this represents a phenomena known as confinement. Effectively, this is a weird quantum mechanical phenomenon that is similar to the fact that you are changing the system by making a measurement. What confinement says is that if you’re at a low enough energy scale (say 1 MeV for protons), you cannot discern quarks from antiquarks because there is not enough energy to break the so-called “Dirac String” connecting the two particles. However, a lot of important quantum mechanical (and even condensed-matter) effects take place around the “condensation” energy (the energy level where confinement occurs for a certain particle). The best attempt to solve this problem is to use something called Lattice Gauge Theory, where you fix a different gauge from the symmetry group of your theory [ for QCD] at each point and use some nearest-neighbor computations. So what does this have to do with having one universal constant?
Well String Theory effectively says that there is one main scalar field (i.e. a function of space-time that assigns a number to each point of space-time; this seems like a tautologous definition, but scalar fields are actually functions on a configuration space, which is in general different than “space-time” itself.) which automatically determines the coupling constants for different fields by looking at it’s expectation. This field is called the dilaton and the universal (String) coupling constant is defined as , where is the dilaton field [ is the configuration space]. This coupling constant determines the “tension” in a string, just as if one were in classical mechanics. As it turns out, this means that you’ve avoided all the infinities that occur in conventional quantum field theories, because there is a universal coupling constant that doesn’t depend on energy scale. In fact, all of the energy scale dependence is in the fields themselves. This means that a String Theory inherently avoids the somewhat arbitrary algorithms that one follows in Quantum Field Theory to “subtract infinity from infinity.” Think of it as a constructive formulation of a theory from the ground up as opposed to a destructive theory that works from bigger objects down. When you’re trying to “integrate” over a space of bigger objects in a space of lower dimension, you’ll always get an infinity. The converse, however, is similar to the fact that a plane embedded in 3-space has zero volume in 3-space because it’s “width” is zero.
I’m not sure if that answers your question, because the extra dimensions aren’t necessarily the energy source; rather they are required to make sure that Poincaré Invariance (Lorentz Invariance plus Translational Invariance) holds and so that anomalies (i.e. quantum predictions that have expectations that don’t jive with classical mechanics) don’t occur. Perhaps they are an energy source, but it’s hard to elucidate the precise physical meaning of that. Rather, the “tension” in the string gives you the energy and that tension is derived from a 1-dimensional object (a string). The extra dimensions are simply for the string to reside in and not violate Lorentz or Translational invariance. Think of it this way: A spherical point particle (0-dimensional object as the radius ) “requires” 4-dimensions for Lorentz Invariance (3 space, 1 time) and for a string we require 6 extra dimensions to be consistent. In fact the first String Theory (Bosonic String theory) required 26-dimensions, but if one uses supersymmetry you can bring it down to 10-dimensions and include fermions. Just for completeness, let me just explain what supersymmetry is, albeit heuristically. In the real world, we have fermions and bosons and from the Black Body Spectrum and quantum statistical mechanics, we know that they obey different statistical laws. Since statistics are derived from a choice of probability measure on a separable Hilbert Space, this means that we need two Hilbert Spaces, equipped with different integration measures, to accommodate both fermions and bosons. As Dirac realized, fermion expectations rely on anticommutivity of operators whereas bosonic expectations rely on commutivity of operators. This means that our Hilbert Space should split into the direct sum of two Hilbert Spaces, . Think of a direct sum as adding the other Hilbert Space’s basis vectors as independent dimensions, just as is constructed as , where the first contributes the basis vector and the second contributes the basis vector. Great, but what about relationships between and ? This is what supersymmetry assumes: There exists an operator that acts something like the annihilation and creation operators of the harmonic oscillator. This means that I can “trade” a fermion for a boson at some type of energy cost. There is no physical reason for this, but it certainly makes the math a lot easier so physicists took the path of shallowest ascent.
I’m not sure if that really answered his question or not, but I suppose that time independence is directly related to the metric, which plays a role in the development of the function in non-linear space-times.
Leave me a lot of questions