The H(-1/2) Hilbert space

We propose an alternative mathematical framework for the Standard Model of Elementary Particles (SMEP), which replaces gauge theory and variational principles: The underlying concepts of exterior derivatives and tensor algebra are replaced by (distributional) Hilbert scales and (purely Hamiltonean) variational principles. As a consequence, the vacuum energy becomes an intrinsic part of the variational principles, i.e. it is identical for all considered Lagrange resp. Hamiltonian mechanisms of all related differential equations, while the corresponding "force" becomes an observable of the considered (Hamiltonean)minimization problem.

In some problem statements of the YME there are basically two assumptions made (which are not clearly defined):

1. the energy of the vacuum energy is zero

2. all energy states can be thought of as particles in plane-waves.

As a consequence the mass gap is the mass of the lightest particle.

Our challenge of proposition 1 is about the measure of the vacuum energy, which gives the value "zero". While the energy norm in the standard H(1) Hilbert space might be zero, the value of the quantum state with respect to the energy norm of the sub-space H(1/2) still can be >0.

Our challenge of proposition 2 is going the same way: a particle with mass can be measured (condensed energy), i.e. it is an element of the test space H(0), while there still can be "waves" in the closed complementary space H(-1/2)-H(0), where the test space is "just" compactly embedded. Those "waves" might be interpreted as all kinds of today's massless "particles" (neutrinos and photons) with related "dark energy".

As a consequence there is no mass gap, but there is an additional vacuum energy governed by the Heisenberg uncertainty principle.

The central concept is about an alternative harmonic quantum energy model enabling a finite "quantum fluctuation = total energy", while replacing Dirac's Delta function and Schrödinger's momentum operator by orthogonal projections onto (standard) test space H(0).

A physical interpretation could be about "rotating differentials" ("quantum fluctuations"), which corresponds mathematically to Leibniz's concept of monads. The mathematical counterparts are the ideal points (or hyper-real numbers). This leads to non-standard analysis, whereby the number field has same cardinality than the real numbers. It is "just" the Archimedean principle which is no longer valid. This looks like a cheap prize to be paid, especially as hyper-real numbers might provide at least a proper mathematical language for the "Big Bang" initial value "function" and its related Einstein-Hilbert action functional.

Looking on hyper-real numbers from the "real" number perspective one must admit to classify the term "real" is a contraction in itself, if it is understood as real. Already the existence of an irrational number (not only the existence of a transcendental numbers) and also the cardinality of the irrational numbers, which is very much different from the rational numbers) is ensured by an axiom, "only" (the Cauchy convergence criterion), i.e. the "empty space" between two rational numbers is filled with infinite irrational numbers with same cardinality as the field of real numbers itself, i.e. with multiple "universes". The difference of real numbers to hyper-real numbers is "just" the fact that there are additionally infinite small and large numbers "existing", ensured "just" by the missing Archimedean property. This principle is basically nothing else than the property that any finite distance can be measured by a given standard measure (i.e. for any real positive number r and a given standard measure of length a (e.g. a=1) there is an integer n that n*a>r). We emphasis that this is still given in the test space, but no longer valid in the distributional Hilbert space, where L(2) is a closed sub-space of.

We further emphasis that

- the differentiable manifold framework of Einstein's field theory does not allow singularities, as required to model black whole and dark energy phenomena

- the hyper-real /ideal points /monads above map to the "proper and terminal indecomposable past-sets/ideal points in space-time (PIPs and TIPs)" in the context of the cosmological censorship and the existence of past and future time-space singularities (GeR).

- the idea to apply Non-standard Analysis (and its related non-classical motion) to explore a Quantum-relative Universe is not new (PoP).

- the model enables an alternative concept to current symmetry breaking and inflation model for the early universe by which the required energy to generate matter out of photons (w/o violating conservation laws) is released during the symmetry breaking process.

The protagonists of the proposed physical-mathematical modelling concept and its corresponding philosophical consequences / interpretations are E. Schrödinger and H. Weyl. We recall a few of their related papers / books

Schrödinger E.,

My view of the world, OX BOW PRESS, Woodbridge, Connecticut, 1983

The interpretation of quantum mechanics, OX BOW PRESS, Woodbridge, Connecticut, 1995

Mind and Matter, Cambridge University Press, 195

What is life? Cambridge University Press, Cambridge, London, New York, Melbourne,1944                

Nature and the Greeks, Science and Humanism, Cambridge University Press, 1996

Weyl H.,

Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton and Oxford, 1949

Was ist Materie, Verlag Julius Springer, Berlin, 1924

The Continuum, A critical examination of the foundation of analysis, Dover Publications., Inc., New York, 1987

Reine Infinitesimalgeometrie (Purely Infinitesimal Geometry), Mat. Z. 2 (1918), 384-411

Erkenntnis und Besinnung, Studia Philosophica 1954