The Unknown Conservation Laws

There are six standard conservation laws of physics: energy, momentum, angular momentum, charge, baryon number and lepton number. It is not generally recognized that there are also a vast number of other conservation laws in physics which are rigorously conserved and quite independent of these six. A simple proof of these other conservation laws is given as well as examples. The implication of these additional conservation laws is discussed for elementary particles.


Introduction
There are six currently recognized conservation laws in physics.The first two (energy and momentum or simply 4-momentum) are well verified and elegantly included into General Relativity as the stress-energy-momentum tensor T µν [1].The third conservation law, angular momentum, has been experimentally confirmed from nuc- lear to atomic to cosmological scales.
The fourth conservation law, charge conservation, is experimentally observed, but it shares a unique position among all six conservation laws, because it can be derived directly without any other assumptions once the E & M field tensor is determined.Thus this conservation law is automatically created by the structure of modern physics.
The fifth and sixth conservation laws (baryon number and lepton number) are purely based on experimental observations of elementary particle interactions and lead to conclusions such as: "The proton does not decay since it is the lowest mass baryon."Sensitive tests for proton decay have agreed with this conclusion [2].

Charge Conservation Is Unique
We mentioned above that charge conservation is not just an experimental conservation equation like energy or momentum or baryon number, but can be itself derived from the basic physical theory.We summarize that de-rivation here for clarity because it leads directly to many more conservation laws.
The key principle is that the current density is derived as the divergence of an antisymmetric tensor.That tensor is called the electromagnetic field tensor F µν shown in Equation (1), where E and B are the familiar electric and magnetic fields.
The charge density J µ is then defined as the divergence of F µν , as shown in Equation ( 2).
To show that J µ is a conserved quantity, we again take its divergence in Equation (3) and recognize that since the E & M field tensor F µν is antisymmetric, we can switch the indices of F µν by adding a minus sign in Equation (3).Then we use the commutation of partial derivatives to switch the µ and ν raised derivatives, and we return to minus the divergence of the current density.
If the divergence of the current density equals its own negative, then it equals zero as summarized in Equation (4).This is the definition of a conserved quantity, when we can define a 4-vector such as J µ whose divergence vanishes.The conserved quantity is the time component 0 J (usually denoted by ρ and called charge density) summarized in Equation (5), and the vector component shows the flow of charge density.
Conserved charge density : Equation ( 6) shows a more familiar form of the current conservation law.It says that the time derivative of the charge density matches the divergence of the current density.Here we switch to the more familiar raised index so that the derivatives are simply the normal derivatives.Here 0 is the time component of J (charge density ρ ) and 1, ,3 k =  , the three spatial coordinates for the current density

Now Generalize
Since charge conservation can be derived from the basic theory of electromagnetism, we call charge conservation an "automatic conservation law".It comes about simply because current density is the divergence of an antisymmetric tensor.Despite being automatic, it has been extremely useful in physics.But now we can get another conservation law for an entirely different type of charge by creating any other antisymmetric second rank tensor.Suppose we contract the E & M field tensor with itself to get a scalar term ( ) , invariant in all coordinate systems as shown in Equation (7).( ) F ρ is not just a scaled version of the standard current density ρ , but weights the current density by the scalar an elementary particle because it is a well isolated system of non-zero fields.Since charge is globally conserved, the isolation of each elementary particle ensures that each particle has well defined charges and that every interaction between elementary particles conserves total charge.Now, however, we see each elementary particle does not have just one charge but many more additional hypercharges associated with it.Since the internal field structure of a proton is very different from an electron, we would assume that the higher order charges would also be quite different.This factor alone could prevent a proton from decaying into a muon or a positron unless some third particle could carry away the exact difference in hypercharge between them.
We conclude baryon and lepton conservation have the qualities that we would expect from hypercharge conservation.The do not come from any primary principle except that they are experimentally observed not to decay out of their families.If each type of particle (lepton, baryon for two) has its own unique spectrum of hypercharge, then one type cannot decay into another type unless there is a third particle (or set of particles) that can carry away the exact difference in hypercharges.The observation that higher mass baryons do decay indicates that the family of baryons share a unique and common internal hypercharge structure.The same will be true for the family of leptons.

Summary
We have shown rigorously that our current models of physics predict a vast set of different conserved charges, called hypercharges, which couple together every known field.We have concentrated more on the E & M hypercharges here, because they are more familiar, but there will also be strong and weak hypercharges.We conclude that elementary particles are isolated islands of fields-strong, weak and electromagnetic-each with their own charge and hypercharges, which must all be conserved under every type of interaction.
Since there appear to be so many degrees of freedom that must be conserved in any decay of elementary particles, this level of constraint suggests that there may be forms of radiation that can carry away excess hypercharge to enable some of the decays and interactions to proceed.The other option is that only a few profiles of conserved charges are stable and become what we call particles.Since each family has such a different hypercharge structure from the other families, one type cannot change into the other without some process that adjusts the internal charge profiles.
We suggest that baryon conservation and lepton conservation are simply two more automatic charge conservation laws involving fields with shorter ranges than electromagnetism.
usual E & M field tensor by the scalar field F , and we have a new antisymmetric E & M tensor.It too will generate its own conserved charge we call F ρ , quite different from the usual charge J 0 as shown in Equation (8).