In this and the next three sections (Localization, Small Parts, and No Evidence for Particles), we will explore the possibility of particle interpretations of quantum mechanics. We find there is no evidence for particles. In this section, we will show that the properties of mass, energy, momentum, spin and charge, which are attributed to particles in classical physics, can actually be shown to be properties of the wave function of quantum mechanics. The method used is the mathematical discipline of group representation theory, which I will endeavor to explain in a way suitable for non-specialists.

Physical and mathematical invariance. Ignoring gravity, we don’t expect the results of experiments to depend either upon the orientation (east-west, north-south) or the position of the experimental apparatus. This constitutes physical rotational and translational invariance (no variation in the results for changes in orientation or position).
The equations governing physical phenomena should reflect this invariance. Suppose we take the Schrödinger equation for the hydrogen atom. Then it should be (and is) independent of how the xyz coordinates are oriented. So the equation is invariant under any rotation of the coordinate system (that is, the form of the equation is the same no matter whether you express it in terms of the original coordinates, xyz, or in terms of the rotated coordinates, x´y´z´). The set of all possible rotations, along with the rules for how two successive rotations give a third, constitute a mathematical entity called the three-dimensional rotation group.

The hydrogen atom and the invariance chain. Because the equation for the hydrogen atom wave function is invariant under (the group of) rotations, the solutions to the equation can be classified or labeled according to their angular momentum, or spin as it is usually called. But this is not just a mathematical classification scheme. If a Stern-Gerlach experiment is done on the atom, one actually gets different, measurable results for the different spins. Thus we have this interesting chain—from physical invariance (rotations shouldn’t matter in the outcome) to mathematical invariance to a mathematical classification scheme for different wave functions to an actual, physical, measurable property of the wave function.

The full set of physical invariances. Physical laws are not just invariant under three-dimensional rotations. They are also invariant under relativistic ‘rotations’ in the four dimensions of space and time. Further, they are invariant under translations in space and time.

There is also one more kind of invariance, that of the ‘internal’ symmetry group. Physicists had noticed that there are certain regularities in the properties of the many ‘elementary’ particles discovered. It has been found that these regularities correspond to invariance under a set of non-space-time rotations—for example rotations in a complex three dimensional space (SU(3)) for quarks or perhaps a complex five or six dimensional space if electrons and neutrinos are included.

Mass, energy, momentum, spin and charge. As in the hydrogen atom case, the physical invariances lead, through the above chain of reasoning, to physical, measurable properties of the solutions to the invariant equations. These properties are mass, energy, momentum and spin for the space-time group, and the strong, electromagnetic and weak charges for the internal symmetry group. Thus all the properties—mass, energy, momentum, spin and charge—that we attributed to particles in classical physics are actually seen to be properties of the quantum mechanical wave function!!

[P11] Linearity and the physical invariance properties—relativistic rotations, translations, internal symmetries—imply the particle-like properties of mass, energy, momentum, spin and charge are properties of the wave function.

[P12] Linearity and the invariance properties also imply that the usual conservation laws and laws of addition for energy, momentum, spin and charge hold in quantum mechanics.

One might object that the quantities derived from group representation theory actually refer back to properties of the particles ‘associated with’ the wave functions. But all the mathematical apparatus used above referred only to the wave functions. There is no reason, at least on this account, to add an extra concept (particles) to the structure of physical existence when that concept never occurred in our deliberations.

Significance of this result I. It is truly astonishing that the existence of the centuries-old classical properties of particles, mass, …, charge, is derived in quantum mechanics.
The inputs to this derivation are the invariance principles and the linearity of the quantum mechanical equations.

[P12A] The fact that the consequences of these inputs give the known properties of particles is probably as close as one can come to a proof that invariance and linearity are absolute principles in the description of physical existence.

Significance of this result II. The fact that the classical particle-like properties are properties of the wave functions severely undermines the rationale for assuming the existence of particles.

Significance of this result III. The physical particle-like states of quantum mechanics, in ket notation, are written as m.E,p,S,s_z,Q(mass, energy, momentum, spin, z component of spin, and the three charges). That is, all the labels on states are group representational quantities. Further, the antisymmetry of fermions and the symmetric statistics of bosons are also group representational properties (associated with the permutation group). This suggests that quantum mechanics as we know it is the representational form of an underlying, pre-representational, linear, appropriately invariant theory (see Ref. 4, Group Representational Clues to a Theory Underlying Quantum Mechanics).