Abstract:
Supersymmetry or ad hoc methods such as renormalization are often used to tame infinities that result from divergent functions in quantum physics. Although SUSY particles have yet to be discovered and may be too massive to fulfill their purpose, and, renormalization seems to lack mathematical rigor. Here we offer an alternative method that employs the least-action and Heisenberg uncertainty principles.
Imagine a Lagrangian with divergent terms. One strategy is to renormalize it. Simply discard the divergent terms, especially if they are infinite. However, Paul Dirac had this to say about such methods: "I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!"
Another strategy is to add superpartners that each have the same mass as their respective standard-model counterparts but make an opposite contribution to the Lagrangian. As a result, the divergence vanishes. Albeit, there is a slight problem: the symmetry of Super-symmetry is broken--the superpartners are believed to be more massive than their standard-model partners. This deflates the balloon of vanishing divergences. To make matters worse, there is a complete and total lack of empirical evidence supporting these superpartners.
If renormalization seems like bad math and SUSY particles are nowhere to be found, what other options are there? How about the least-action and Heisenberg uncertainty principles? Let's first examine the least-action principle:
A particle typically takes the shortest path possible between two points. For that to happen, delta-s, at equation 1, cannot be a large, divergent quantity. It should be zero units of action or time multiplied by energy. However, the following is true:
Line 3 shows that time multiplied by energy is greater than or equal to h-bar. To get delta-s to equal zero requires steps 4 through 6:
At 7 we set up another substitution. The final equations are 8 and 9 below:
Equations 8 and 9 show why there's a least action principle and why energy is generally conserved. Suppose we have a conserved energy L. The divergent energy, delta-E, can be interpreted as energy borrowed from the vacuum. Because it's borrowed, it must vanish within time delta-t. The larger this energy, the shorter its lifespan. As a result, the energy L that you start with is the energy you end up with. It is conserved. Also, the action is the least action.
At equation 10 we have a Lagrangian where there is no borrowed energy. Because no energy is borrowed, time delta-t is infinite. In other words, this scenario can last indefinitely and create the impression that energy is always conserved.
At equation 11 we have the opposite extreme: a Lagrangian that diverges to infinity. The good news is delta-t is zero, which shows that infinite borrowed energy does not exist. We can also infer that large borrowed energies exist for too short of a time to be meaningfully observed and measured, so the energy we do observe and measure is small by comparison. Thus, renormalization works despite its ad hoc nature because nature wipes out divergences by means of the uncertainty principle and least action. The only time it is appropriate to keep the divergent terms is when divergent energy is added to the system and not borrowed from nothing.
Now, let's suppose L is a Lagrangian for vacuum energy (see equation 12). A Higgs boson (m-sub-H) pops into existence and has a lifespan of t-sub-H. A too-large Higgs mass would have a lifespan too short to provide a meaningful opportunity to observe it, so the mass we are most likely to observe is a smaller mass.
More examples: Equation 13 below takes into account multiple particles. Equation 14 takes into account a Lagrangian or function with multiple terms and parameters.
Since delta-s must be zero to minimize the action, then delta-s along D dimensions must also be zero. Further, both delta-s and s have units of momentum multiplied by position. If we integrate over position and/or momentum space, the following must be true:
The uncertainty of knowing a particle's position is cancelled by knowing its momentum and vice versa. As a result, the particle's action is minimized along with its position path and momentum.
In conclusion, divergences are tamed if the least-action and uncertainty principles are applied. SUSY particles are not needed and ad hoc methods such as renormalization can be set aside.
References:
1. Lincoln, Don. 2013-05-21. What is Supersymmetry? Fermilab.
2. Martin, Stephen P. 1997. A Supersymmetry Primer. Perspectives on Supersymmetry. Advanced Series on Directions in High Energy Physics. Vol. 18.
3. Susskind, Leonard. 2012. Supersymmetry and Grand Unification Lectures. Stanford University
4. McMahon, David. 2008. Quantum Field Theory Demystified. McGraw Hill
5. Baez, John. 11/14/2006. Renormalizability. math.ucr.edu
6. Renormalization. Wikipedia
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