Application of Quantized Metrics in the Treatment of Non-local Events as Produced in Double slit Experiments

29 01 2013

Gregory C. Matthews

January  27, 2013 (copyright)


This article provides an ontological interpretation of the results of standard double-slit experiments by consideration of formulations of quantized metrics. A quantized metric form describing particle propagation in a quantized space-form would result in non-local trajectories. A reexamination of the double slit experiments may confirm and describe experimentally the exact form of a quantized metric.

This essay explores a reinterpretation of the famous “double slit experiment” in the context of proposed formulations of quantized metrics, in part, to determine if the results of the experiments agree with the formulations of quantized metrics, and, in part, to advance ontological interpretations of the results. The essay argues that a quantized form of the metric is needed in order to model quantum gravity, and that any form of the metric would also yield ‘non-local effects.”

In standard Einstein models of the space-time metric, a fundamental precept of relativistic theories is that theories resulting in non-local effects are invalid, [1], where such phenomena are described by classical relativity theorists as “spooky action at a distance.” Most of relativistic theory was formulated before the advent of quantum mechanics and the formalism adopted in the Copenhagen Interpretation of 1929.

Much of the observed behavior of particles described by quantum mechanics seemed to defy the general intuition of physicists and mathematicians trained in classical and high energy relativistic physics. One of the most famous examples of particle behavior that seemingly violates basic models of space-time constructs is the double slit experiment [2], [3], [4]. In this experiment, single particles are admitted to an apparatus constructed with two slits available for the impingent particle to pass through, and then impinge on a detector. Observations of the pattern produced by the detector apparatus using many repeats of the experiment, show an interference pattern scatter consistent with a model where the impingent particle interferes with itself as it propagates through the double slit apparatus. The observed particle behavior may be seen as a violation of the predictions of classical and relativistic physics. The results of the double slit experiment seem to be consistent with a single particle propagating through both spatially separated slits at the same time.

To this point, most methodologies applied in experimental work with quantum field mechanics have relied on classical measures, such as classical clocks and rulers. To date, observations made using the traditional apparatus of the double slit experiment have relied on classical clocks and rulers, instead of the possibility of using quantized clocks and rulers [5]. With a conjectured quantized metric, the results of the double slit experiment may be interpreted differently.

A number of theoretical physics conjecture have been proposed to explain this apparent “non- local’ behavior of quantized particles. An important aspect of quantum theory is the proposed treatment of quantized forms of the metric as described in [6], [7].

We postulate a quantized form of the metric as: gμν = ΦμΦν (1)

As part of this conjecture, we consider Φμ and Φν as waveforms of the vacuum of the constructed experimental apparatus at the loci of the double slits before impingement of a Dirac particle to the apparatus. We also postulate that the vacua waveforms are not subject to the Pauli Exclusion Principle, and may share identical sets of quantum numbers.

A logical deduction that may be made, given these two assumptions, is that if two or more vacua share the same set of quantum numbers, then, from the perspective of classical space- time, they would have the property of seeming to occupy the same event locus as each other from the perspective of the impingement Dirac particle. The double slit apparatus itself is a form of a classical ruler used in determining classical measures of event loci. From the perspective of a quantized model of the metric, however, a set of vacua particles, including shared identical sets of quantum numbers, at the loci of the two slits would treat the apparent classical spatial- temporal separation of the two slits as a single event locus. We then apply the Calculus of Variations to the formulation of the expected propagation of Dirac particles admitted to the experimental apparatus. An assumption is applied that vacua may share identical sets of quantum numbers in the neighborhood of two classically separated loci corresponding to the two slits of the apparatus, as measured using classical rulers and clocks. Equation (2) models the wave-forms to be considered in application of the Calculus of Variations:

Φ(i)μ = ΦD(i) μ + Φv(i) (2)

Φ(i)μ ΦD(s1)μ + Φv(s1)μ = ΦD(s2)μ + Φv(s2)μ = Φ(s)μ, (3) Setting Φ(s1)Vμ = Φ(s2)Vμ, for vacua sharing identical sets of quantum numbers, and applying

Boolean algebra, A + A = A, for the vacua, we obtain: Φ(s)μ  Φ(f)μ = ΦD(f)μ + Φd(f)μ at the detector (4)

In Equations (2) to (4), ΦDμ represents the wave-form of a Dirac particle, Φv the wave-form of vacua particle. The use of notation of the form Φ(i), Φv(s), and Φv(f) denotes the terms of the waveform of the particles prior to interaction with the slits, the terms at particle interaction with the slits, and the terms for interaction with the detector, respectively. Φv(s1)μ and Φv(s2)μ represent the wave-form of virtual particles in the neighborhood of slits 1 and 2, which may share identical sets of quantum numbers. The term, Φd(f)μ, represents the wave-form of the detector.

Since, Φv (s1)μ and Φv (s2)μ are stated to represent the wave-form descriptor of the two slits, they are indistinguishable from each other as a consequence of sharing identical sets of quantum numbers. Included in this conjecture is an assumption, as suggested in [8], that all particle interactions may be treated as an observation inasmuch as particle interactions result in collapse of the superposed state, whether or not an experimental physicist observes the interaction. In this model, a distinction is made between observations made by bosonic detectors as opposed to fermionic detectors, the latter of which is subject to the Pauli Exclusion Principle.

Fermionic detectors are used in the classic description of the two-slit experiment. As a consequence, the interaction of the impingement particle on the detector behind the two-slits is entirely local since the detector is fermionic, and subject to the Pauli Exclusion Principle.1 No two particles of the detector may share the same set of quantum numbers, and therefore, the particle interaction with the detector is unique to the interaction with the fermionic particle of the detector on impingement.

This is unlike the proposed bosonic descriptor of the vacua constituting the two-slits of the apparatus where the vacua of both slits may share identical quantum numbers. At the quantum scale, both slits may be treated as identical, and hence the particle impingent on the two slits interacts equally with both.

At the point of observation at the detector, however, the impingent particle will contain the total information of a particle that had interacted with both slits. An observation of the contained information of the impingement particle will be resolved at the fermionic detector as if the impingent particle interfered with itself. This set of equations predicts the interference pattern observed generally in the double slit experiments.

Solutions of the application of the Calculus of Variations in this model would predict that an impinging Dirac particle would propagate through both slits where the quantum numbers of the vacua in the neighborhood of the two slits were identical, as if, from a frame of reference of quantized rulers and clocks, the event loci of the two slits were identical. Solutions may be made by inspection.

In summary, by application of a postulated form of a quantized metric to the modeling of a double slit experiment, it may be stated that the event loci would have the appearance of non- local action at the two slits, where the vacua are not subject to the Pauli Exclusion Principle.

Such results may explain apparent contradictions with Einstein’s prohibitions on non-locality in cases where quantized metrics are conjectured as opposed to metrics based on classical clocks and rulers constructed of arrays of fermions. Similar models of the vacua may be developed to describe the electron cloud geometries.


1. Einstein, A, Relativity, the Special and the General Theory, Crown, New York, 1961.

2. Feynman, Richard P. (1988). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0-691-02417-0.

3. Pfleegor, R. L. and Mandel, L. (July 1967). “Interference of Independent Photon Beams”. Phys. Rev. 159 (5): 1084–1088. Bibcode 1967PhRv..159.1084P. doi:10.1103/PhysRev.159.1084.


1 A postulate is made that local behavior is required for fermions as a function of the Pauli Exclusion Principle. Non-local behaviors would require a violation of the Exclusion Principle, a principle not applicable to bosons.

G. C. Matthews, January 27, 2013


5. G. C. Matthews, A Note on Einstein’s Twin Paradox Applied to the Behavior of Entangled Particles, unpublished, February 1, 2011.

6. G. C. Matthews, A Proposed Treatment of Quantum Vector Field Theory, unpublished, April 18, 2012.

7. G. C. Matthews, A Proposed Treatment of the Vacuum as a Field of Bosonic Virtual Particles, October 21, 2011.

8. G. C. Matthews, Bayes Theorem Applied to Scalar Wavefunctions and the Statistical Derivation of the Second Law of Thermodynamics, unpublished, November 22,2008.




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