Supersymmetry, Axial Anomaly and Holomorphy: a problem and a solution.

Reason, or the ratio of all we have already known, is not the same that it shall be when we know more.~ William Blake!
The essential aspect of supersymmetry (SuSy), its bosonic-fermionic symmetry, also poses the gravest problem for the theory: since bose-fermi symmetry is not observed in ‘nature’, it must be broken if SuSy is to make contact with reality. A promising possibility is spontaneous SuSy-breaking that occurs at tree-approximation level induced by scalar multiplet interaction: this leads to zero-mass Goldstone fermions; that however cannot be identified with the neutrino: a solution naturally presents itself, namely, involving the Ward identities of the theory, which connect the Green’s functions of the supercurrent to other matrix elements: but the validity of the Ward identities require that no anomalies of the supercurrent exist. The reason this is so important is that axial currents create via creation-annihilation Hilbert Space operators, in the context of SuSy, the graviton with all the properties needed for unification to occur between General Relativity and Quantum Field Theory. Such gravitonic axial-current creation is a consequence of the convolution of the SuSy group generators and the generators of the PoincarĂ© group: it is in this sense, along many others, that SuperString Theory is essentially a quantum theory of gravity. However, one must show that axial anomaly cancellation can be achieved, for two reasons: 1) axionic (super)-current gravitonic creation is one of the central reasons why SuperString/M-Theory is finite (it needs no renormalization) and thus is a quantum theory of gravity, and 2) is the only theory that ‘bypasses‘ (avoids the loopholes of) the severely limitative Coleman–Mandula theorem! the question now arises, in the crucial Yang-Mills context: is axionic anomally cancellation possible and is it paradoxes-free? I will show that one can solve both problems. But first: why are General Relativity (GR) and Quantum field Theory (QFT) so fundamentally incompatible? Here are just a few reasons:

GR is in an essential sense a theory of the gravitational field represented by the pseudo-Riemannian metric of an ‘inseparable’ spacetime. Since QFT quantizes all fields in spacetime, the gravitational field also must be quantized. Let us look at the field equation of GR:

    \[{R_{\mu \nu }} - \frac{1}{2}\,{g_{\mu \nu }}\,R + {g_{\mu \nu }}\Lambda = \frac{{8\pi G}}{{{c^4}}}{T_{\mu \nu }}\]

The side of the GR field equation that describes matter sources are under the descriptive domain of QFT, while the other side describes gravitation as a classical field: since a quantization of gravity implies that spacetime is not ‘smooth’ and since integration is a smoothing process, the GR-action principle would have no solution; and, since the action contains the content of a physical theory, that means that GR does not even ‘describe’ spacetime. So, if the right hand side of the GR-field equation represent quantized matter, GR can be shown inconsistent since consistency presupposes the existence of solutions.

Moreover, the gravitational field is represented by the spacetime metric, and a quantization of the gravitational field is a quantization of that metric: the problem now is that the quantum dynamical description of the gravitational field of GR implies a dynamical quantum-spacetime, but QFT presupposes a fixed non-dynamical background spacetime for the description of all fields. Thus, a quantum theory of the GR-gravitational field cannot be independent as a QFT because the ‘active’ diffeomorphism invariances of GR are ontologically incompatible with any fixed background spacetime!

Also and more serious, in the Hilbert space setting, energy and time are ‘entangled’ in the Heisenberg Uncertainty Principle: let


stand in for that relation. If QFT treats time as it must, as a global