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Structure of a spin 1/2

The non-hermitian states that lead to separation of the four Bell states are examined. In the absence of interactions, a new quantum state of spin magnitude 1/(root(2) is predicted. Properties of these states show that an isolated spin is a resonance state with zero net angular momentum, consistent with a point particle, and each resonance corresponds to a degenerate but well defined structure. By averaging and de-coherence these structures are shown to form ensembles which are consistent with the usual quantum description of a spin.


INTRODUCTION
In spite of its tremendous success in describing the properties of microscopic systems, the debate over whether quantum mechanics is the most fundamental theory has been going on since its inception 1 . The basic property of quantum mechanics that belies it as the most fundamental theory is its statistical nature 2 . The history is well known: EPR 3 showed that two non-commuting operators are simultaneously elements of physical reality, thereby concluding that quantum mechanics is incomplete, albeit they assumed locality. Bohr 4 replied with complementarity, arguing for completeness; thirty years later Bell 5 , questioned the locality assumption 6 , and the conclusion drawn that any deeper theory than quantum mechanics must be non-local. In subsequent years the idea that entangled states can persist to space-like separations became well accepted 7 .
However the EPR coincidence experiments on entangled photons 8,9 that demonstrate violation of Bell's Inequalities cannot be rationalized within quantum mechanics but are purported to depend on non-local connectivity between the pair 10 . Non-locality is a concept, as yet, not understood 11 .
If changes are made to quantum theory which can explain notions like non-locality, then these must simply resolve quantum theory into something deeper and restore it again in an appropriate limit. Replacing such a successful theory is not the goal. In this paper the treatment involves spin ½ only. It has been shown that the Bell states 12,13 , maximally entangled within quantum theory, separate into products of single particle states when a specific non-hermiticity is introduced. Motivated by this property, these non-hermitian states are shown to be consistent with both EPR's requirement of reality 3 and, after ensemble averaging, also quantum theory, which suppresses or treats the information associated with the non-hermiticity as random.
The consequences of the changes introduced here are mostly benign. The non-hermitian state operators retain trace 1 and are idempotent, consistent with pure states. They have real eigenvalues, which are the same as those when the non-hermiticity is dropped. The only mathematical difference is the two eigenstates are non-orthogonal.
The most significant change is in the interpretation of the non-hermitian states. Rather than being the statistical operators of quantum theory, they are treated here as representing the pure spin states of a single particle. This is based upon the premise that single particles exist and therefore there must be a way of representing their states. Silver atoms, oxygen molecules, electrons etc. are the individual components that make up systems containing enough particles for reliable and reproducible measurements. In analogy with the ideas of classical statistical mechanics, the ensembles that form the quantum states are proposed to be composed of individual particles whose states are non-hermitian.
In the following it is first argued that the non-hermitian spin states suggest an underlying physical structure for a spin ½. Using common requirements of quantum theory, the states of an isolated spin are found to be 8-fold degenerate with eigenvalues of ±1. The view that emerges is one of resonance between these degenerate spin states that is not inconsistent with the quantum conclusion that an isolated spin acts like a point particle 14 . When such spins are influenced by local or external magnetic fields, however, all states randomize except those quantized along the field axis, leading to the usual view of a spin ½ as, for example, filtered in a Stern-Gerlach experiment. In addition, it is shown that phase randomization of an ensemble of non-hermitian spins leads to the usual hermitian spin states of quantum theory. In fact one conclusion is the non-hermiticity describes a phase, and only in unique situations (e.g. apparently to date only coincidence experiments on photons) can it be detected.  In contrast the non-hermitain states can only be written in off-diagonal form s    . Therefore it is necessary to treat the non-hermtitian states as operators in Hilbert-Schmidt space with Hilbert-Schmidt norm 19 . The time dependence is then given by the quantum Liouville equation rather than the Schrödinger equation.
The measurement postulate remains the same as ˆT r A A       . The relationship between the non-hermitian states and the density operator involves ensemble averaging. In the case of the singlet, its isotropy renders it invariant in 3D space and therefore the average over all different orientations, r , gives,   12 12 12 12 EAˆŝ p s d s where the bar denotes ensemble averaging (EA) and   p r is a distribution function. Further discussion of the Bell states is deferred to a later publication 13 . In the following, the non-hermitian states that separate the Bell states are investigated.

PROPERTIES OF THE NON-HERMITIAN STATES
As prepared for experiment, and after many particles have passed a Stern-Gerlach filter, the eigenvalues of Z  are +1 and -1 with orthogonal quantum states in the laboratory frame of Z Z 1 0 for 1 and for -1 Although from a random source on the average, fifty percent of the spins are deflected up and fifty percent down, there is no possibility within quantum theory to predict a priori which way a spin will be deflected, or which slit an electron passes in a double slit experiment. These are examples of the statistical nature of quantum theory.
In contrast the non-hermitian states of the single events before reaching the Stern-Gerlach filters are deterministic by the definition Eq.(2.4), but this is usually masked by the degeneracy associated with them. There are eight different choices of integers in Eq.(2.4) each giving the same eigenvalues and displaying states that differ from the others only in sign. Whereas the eigenvalues of Z are ±1 with eigenstates give by Eq.(3.1), the non-hermitian spin operator in Eq.(2.4) have eight different forms due to the permutation of integers   , , z x y n n n . Each choice corresponds to one of the octants in the frame, ˆˆx, y, z . One of these is and has the same eigenvalues of ±1 but with non-orthogonal eigenstates, One eigenvalue is associated with theẑ axis and the other with thex axis. For all choices of the integers in Eq.(2.4), the eigenvalues are always +1 and -1 and the associated eigenstates pairs for each octant are the various nonorthogonal combinations of z z z z 1 1 1 1 ; ; That is the eigenvalues of ±1 are eight-fold degenerate. There are no eigenstates associated with y component. The view that emerges is consistent with the way particles with structure are treated in, for example, spectroscopy. Consider a symmetric top molecule which has internal angular momentum which can be observed spectroscopically from the laboratory frame. The usual technique is to define a body fixed coordinate frame, ˆˆx, y, z , and express the internal properties in terms of, in the case of rotations, angular momentum around two axes due to it having two different moments of inertia. These are generally called the rotational angular momentum J and the projection of this on its symmetry axis, K. Once these are defined, an ensemble of such particles, all with different body frames, is transformed to the laboratory frame where they are observed statistically.
At any instant, a spin can exist in only one of its eight octants, but all are assumed to be equally probable in the absence of interactions. That is, whereas a single spin is oriented and well defined in one octant, it is impossible, due to degeneracy, to distinguish it from others that are oriented in different octants of the same body frame,  ˆx,y, z .
The structure of a spin, as seen from Eq.(2.4), displays two angular momentum, one associated with the body frame ẑ axis, and the other with the x axis, see also Eq. (3.4). This leads to a two dimension structure for each of the eight orientations within the body frame, see Figure 1. The non-hermiticity is a result of the term y i which is not interpreted here as angular momentum but rather as a rotation operator. It orients the 2D spin in 3D space via the quantum equivalent of the cross products, z x y i     , see Figure 1.

The hermitian resonance states of a single spin ½.
A basic tenet that emerges from the development here is the non-hermiticity, needed for its physical reality of spin and the separability of entangled states, gives way to hermitian states except in special cases. One such case is seen from Eq.(2.5) where the singlet is composed of pairs of non-hermitian states even though these combine to give the singlet, which is hermitian. Likewise, as shown in below, when in the presence of a magnetic interaction, the nonhermiticity is phase randomized. In this subsection it is shown, due to the quantum requirement of indistinguishablility, the states for an isolated spin are hermitian.
Analogy can be made with the description of chemical bonds, such as the hydrogen molecule. In the early treatments, three contributions to the bond strength were identified being the overlap integral, the Coulomb integral and the exchange integral. The last arises due to the indistinguishablility of the two arrangements, A Geometrically these eigenstates lie at angles of 45 o between the four sets of the two axes defined by ẑ n z and x n x , Eq.(3.7), and are a superposition of the non-orthogonal states in Eq. (3.4). The resulting spin of magnitude √2 is a pure quantum resonance in origin. In terms of the usual spin magnitude / 2  , the operator for the above spin states for a single particle is defined by, If the environment is not isotropic, such as when the spin interacts with local or external magnetic fields, the two axes become distinguishable and the √2 resonance state cannot be formed. The spin, Eq.(3.11) is predicted to exist in an isotropic environment only, in the absence of all interactions.
The following three subsections show that the non-hermitian states give a consistent description of isolated spins; spins in a bulk isotropic system; and in the presence of an external magnetic field. In all cases, the ensemble averaging and de-coherence renders to states hermitian.

An isolated spin
In the previous section the resonance state can exists only for an isolated spin oriented in any one of the quadrants of its body frame defined by the ˆx, z plane, see Figure 1. However there is no discernable difference between any of these four orientations so that the state of an isolated spin is given as an equal superposition of the four, The view here is different from the classical meaning of objective reality. Here a single spin is viewed as being in a well defined state described by Eq.(3.6) at an instant of time, and those states are dispersion free. However equally probable is its orientation in the three other quadrants. Since the sum of the four resonances gives the identity, and a state of zero net angular momentum, the point particle description of intrinsic angular momentum is corroborated.
Resonance is a common property of quantum systems. By definition, resonance means that there are a number of equivalent structures of a system and at any instant only one of these exists. Common chemical techniques on molecules make use of resonance to promote the reactivity of one structure over the others. Likewise, the resonance state for the 2D spin has the same meaning. At any instant the spin can be in any one of its eight degenerate states. This is consistent with the inability of quantum theory to predict with certainty which resonance state a spin actually occupies. In the vast majority of cases such information is irrelevant and so, in deference to its success, is lost in the statistical nature of quantum theory.

De-coherence in the laboratory frame.
In this subsection the spins oriented in their body frame are transformed to the laboratory frame where it is shown that de-coherence renders the macroscopic states hermitian and leads to consistency with the usual results from quantum theory.
The laboratory frame, ˆˆX , Y, Z , is chosen randomly in any convenient direction in isotropic 3D space. The body and the laboratory frames differ by a rotation by Error! Bookmark not defined. ,   The states in the body and laboratory frames are related by  3.14) and the Pauli spin components between the two frames take the form lost giving rise to entanglement and dispersion. This is the price paid for retaining hermitian ensemble averages over the non-hermitian states, but is also a strength of quantum theory which suppresses information which is usually random. The notion of objective reality is different here from the classical definition. If it were possible to perform reproducible experiments on a single particle which did not disturb it, then the three operators describing a spin ½,   , , z x i    must all exist simultaneously. Quantum effects; resonances; and de-coherence at the quantum level can randomize the information due to the non-hermiticity.
Extension to existing theories should lead to new predictions. The major prediction here is the existence of the √2 spin. The only experiments known to date which are taken as support for non-local connectivity through violation of Bell's Inequalities, are the various coincidence experiments done with photons. Although quantum theory correctly predicts the correlation between entangled pairs, it does not give a basis for understanding why various filter settings lead to those violations. In a later publication it is shown that the presence of the √2 spin is responsible for such settings, and can also rationalize the violation of the CHSH from of Bell's Inequalities by 2√2.
It remains to be shown that the other properties of spin are consistent with the 2D structure presented here. These include evaluation of the electron g-factor and application to experiments such as the double slit on electrons which is expected to be dependant on the resonance states of a single spin.