Spacetime may be a mere perspectival model within a universal mind

Reading | Physics

Ben Werner, M.Sc. | 2025-01-10

Introduction

This essay discusses several paradoxes within physical theory that can be resolved with the concept of macroscopic “quantum spacetime”—specifically, complex projective space, which is the geometric space of quantum wave functions. Whereas this argument does not contradict general relativity as a theoretical model of spacetime, if the actual geometry of spacetime is in fact complex projective space, the implication is that non-dual consciousness must be the substratum of the universe, within which a macroscopic quantum wavefunction forms the universal noumena behind individual subjective experiences.

The most familiar of the paradoxes discussed in this essay is quantum “spooky action at a distance.” Each of these paradoxes can be conceptually resolved by adopting a view that Euclidean spacetime—in which we construct our classical concept of the universe from a human perspective—is a model within a macroscopic complex projective space. The statement that Euclidean geometry may be constructed as a model within a more fundamental projective geometry is matter-of-factually true from a geometric-theoretical standpoint [1]. And the statement that the geometric basis of quantum wavefunctions is complex projective space is also matter-of-factually true [2]. However, the core idea proposed here—that complex projective space is the actual spacetime within which the universe exists at all scales, including the human scale—is not an accepted fact. Whereas this idea does not conflict with presently accepted physical theory (because Euclidean spacetime may be constructed as a model within complex projective space) it does contradict the implicit assumption behind the development of physical theory: that a physical reality exists independently of the subjective observer. This contradiction derives from the fact that, if we suppose complex projective space as the actual geometry of spacetime, then Euclidean spacetime can only appear from a point of perspective i.e., as the experience of a subjective observer.

For readers unfamiliar with projective space, a recommended foundational resource is [1], wherein projective, Euclidean, spherical, and hyperbolic geometries are developed alongside each other. Whereas the dimensions of real projective space are orthogonal real number lines, the dimensions of complex projective space are complex numbers, wherein the orthogonality of dimensions is due to the orthogonality between the real and imaginary components of complex numbers. An understanding of (complex) projective space may be developed through several different approaches. Within the context of this essay, the following concepts are particularly meaningful:

• Projective space is the geometrical formalization of a space ‘seen from all perspectives at once.’ Accordingly, Euclidean space can be constructed as a model within projective space, equivalent to taking a single point of perspective within the projective space. Projective space is the more elementary geometry, with fewer axioms than Euclidean space.
• There is no meaning to distance (and hence separation) within projective space. This fact aligns with complex projective space being the geometric basis of quantum wavefunctions (and “spooky action at a distance” due to entanglement). There is no meaningful way to define or visualize separate objects within projective space.
• Complex projective space can be understood as a geometry intrinsic to complex numbers, rather than an amalgamation of number and geometry as with the construction of “orthogonal number lines” in Euclidean geometry. This aligns with a metaphysics where consciousness is fundamental, wherein seemingly ‘unconscious’ ideas can form the basis of structure and physical laws that govern an objective world.

The following sections describe several paradoxes within physical theory—and experiments—that can be conceptually resolved by assuming that the observed universe appears within (macroscopic) complex projective spacetime. These paradoxes have only appeared in physical theory within the last few decades, as a result of the effort to ‘fill in the gaps’ of a grand unified theory, unifying quantum and classical models. The resistance of physical theory to grand unification may be because we have left out something essential, namely, the structure of the conscious space within which the universe appears.

The paradoxes discussed in this essay are:

• The paradox of entanglement of photons over vast distances.
• The paradox of static virtual fields (the fact that electrostatic and magnetostatic fields have an imaginary wave number).
• The paradox of the missing negative mass/energy (the fact that we do not observe negative mass/energy yet are surrounded by it according to physical theory).
• The paradox of instantaneous virtual field propagation (the experimental observation that changes in virtual electromagnetic field components propagate instantaneously).

 

The paradox of entangled photons over vast distances

This paradox (mentioned in the introduction) is often framed by the extreme example of light emitted from a distant galaxy—perhaps billions of lightyears away—which, according to quantum mechanics, might represent entanglement (non-separability) between a system in the distant galaxy and a system on Earth. Within a conceptual model of a universe set within Euclidean space, quantum entanglement seems to imply some kind of instantaneous field propagation connecting the two systems—a possibility excluded by special relativity and the limiting speed of light. Special relativity seems to explain this paradox away with the assertion that, from the frame of the light wave, the universe is flat and hence there is no separation between the two systems. However, the paradox remains as long as we believe that the resting frames of the two systems (modeled as three Euclidean dimensions of space plus a fourth dimension of time) are fundamentally as real as the frame of reference of the light wave.

Conceptual resolution of the paradox: A 2D projective plane may be constructed from a 2D Euclidean plane by co-identifying each pair of antipodal points along the edge of the 2D Euclidean plane. Similarly, we can transform the model of a light wave traveling through 4D Euclidean spacetime into the same light wave within complex projective space by co-identifying the point in time when the light wave is emitted with the point in time when the light wave is absorbed. This aligns with a principle of quantum mechanics which states that the emission and absorption of a light wave is a single event. Accordingly, the paradox of entanglement of photons over vast distances can be conceptually resolved by assuming that the fundamental spacetime of the universe is (macroscopic) complex projective space. In this view, the universe exists as an entangled whole, even while we experience a perceptual model of a universe of separate objects set within Euclidean spacetime. This idea aligns with an instinctual feeling that a ‘universal-now’ exists in a meaningful sense, in spite of our being taught—per special relativity—that such universal-now is not an actual reality. However, there is no conflict between an actual universal-now and special relativity, provided that we clarify that the universal-now applies to an entangled universe of non-separate wavefunctions. If we want an objective experience of a distant galaxy, we must still transmit that experience via light waves and/or travel through space at less than the speed of light to said galaxy. How a person might experience an entangled universe in the universal-now, and whether there is any way to communicate or visualize such an experience, is a separate matter. The implication of the concepts proposed in this essay is that such an experience would be more fundamental to our capacity as conscious entities than reflecting on a mental model of a universe set within 4D Euclidean spacetime.

 

The paradox of static virtual fields

For internal consistency within quantum-electrodynamics, the photons associated with electrostatic and magnetostatic fields are defined as virtual photons, in contrast with the real photons associated with electromagnetic waves [3]. While the wave number of a real photon is always a real number, the wave number of a virtual photon is an imaginary number, implying that, while real photons have real energy, virtual photons (in a certain sense) have imaginary energy. An interpretation that the energy of electrostatic and magnetostatic fields is imaginary presents a paradox, because an observed (real world) quantity can only be imaginary with respect to some other quantity with which it is in relative harmonic motion. For example, a simple pendulum has a kinetic energy and a potential energy, each of which can be measured as a real quantity, just as the energy of electrostatic and magnetostatic fields can be measured as a real quantity. However, if the pendulum is swinging back and forth, its potential and kinetic energy are imaginary with respect to each other. Similar examples can be given for the quantities that describe anything that is rotating, oscillating, or vibrating. Therefore, it is a paradox that electrostatic and magnetostatic fields are static and imaginary. The implication of this paradox is that, if we are measuring a quantity that is static and imaginary, then we as observers, or something about the act of observing, must be characterized by a motion that is rotating, oscillating, or vibrating (in spite of the fact that we are not conscious of such a motion in our normal sensory observation of the world). The notion of a hidden harmonic motion is reinforced by the fact that we are ‘bounded’ by harmonic motion: at the quantum scale, everything is characterized by vibration; at the speed of light, only light waves exist; and at the cosmological scale, there is a beginning and an end to the flat spacetime within which our objective experience happens, perhaps in a repeating fashion, where the end of one universe may be the beginning of another universe [6].

Conceptual resolution of the paradox: The paradox of static virtual fields can be resolved with the concept that the fundamental space in which electrostatic and magnetostatic fields exist is complex projective space, which is itself the space of quantum wavefunctions. An analogy of surfing ocean waves might be helpful to develop this concept. It is easy to see that, if we are riding an ocean wave, the wave will appear static to us, even though it is moving from the perspective of someone standing on the shore. Visualizing static virtual fields within a quantum wavefunction is more difficult, because a wavefunction is not an observable object like an ocean wave. Electromagnetic waves are themselves quantum wavefunctions, and even though we are accustomed to visualizing them as waves moving through space, the invariance of light speed with respect to one’s frame of reference means that we can never observe a light wave as an object. The same fact holds true for quantum wavefunctions describing stationary particles: the wavefunction itself is never observable. However, just as we can register the effect of an electromagnetic wave in the oscillating electric and magnetic fields of an antenna, static electric and magnetic fields can be thought of as registering a macroscopic quantum wavefunction. Whereas the non-observability of a light wave can be attributed to the invariance of light speed, the non-observability of a macroscopic quantum wavefunction could be attributed to the asymmetry in how we experience space versus time: within our Euclidean view of spacetime, we perceive a macroscopic extent of space at any moment, but we only perceive a microscopic extent of time at any moment.

Since a quantum wavefunction encompasses equivalent scales of space and time within complex projective space, we can only measure or perceive the components of such wavefunctions that fit within our Euclidean view of spacetime, which, again, consists of a macroscopic extent of space but a microscopic extent of time. An experience of a ‘complete’ macroscopic quantum wavefunction could only happen in complex projective space, which would be an experience ‘outside’ the passing microscopic moment in Euclidean time. Projective space is defined as space “seen from all perspectives at once,” which would be to say that all points of perspective are entangled (non-separable) within a macroscopic quantum wavefunction. Therefore, such an experience would be coincident with a dissolution of the subject-object division; that is to say, with a shifting of identity away from a single point of perspective. Such a ‘movement’ would not be the movement of something observable like a physical pendulum, but rather a hidden (normally ‘unconscious’) movement intrinsic to the co-existence of Euclidean spacetime (supporting a subject-object division) and complex projective space (in which a subject-object division is impossible). This may be the hidden movement implied by the paradox of static virtual fields.

 

The paradox of missing negative mass/energy

Negative mass/energy is sometimes referred to as “exotic” because it is never directly observed, even though its existence is necessitated by our physical models. At the cosmic scale, the most current measurements of the expansion rate of the universe tell us that the universe is essentially flat [7], meaning that the positive spacetime curvature correlated with observable positive mass/energy must be balanced by a negative spacetime curvature of an unobserved negative mass/energy. The overall shape of the universe, as the whole of spacetime, seems to be a balance of positive and negative curvature. Furthermore, since photons and anti-photons are the same thing, anything moving at lightspeed is balanced positive and negative energy. And at the smallest scale—the Planck scale—quantum theory predicts that the vast majority of energy present in quantum vacuum fluctuations are virtual particles, which are intermediaries between positive and negative solutions to the energy-momentum relation. The paradox, therefore, is that the universe appears to be fundamentally a balance of positive and negative mass/energy, and yet we only observe positive mass/energy at the human scale. We seem to be literally ‘bounded’ by a balance of positive and negative mass/energy at the micro and macro limits of the universe, and at the limit of light speed.

Conceptual resolution of the paradox: The non-observability of quantum wavefunctions can account for why we do not observe negative mass/energy directly at the human scale, in spite of the ubiquity of negative mass/energy in counterbalance to positive mass/energy at the boundaries of our objective experience. In order to frame the meaning of negative mass/energy within physical theory, we can turn to the recent synthesis of information theory and physical theory, which indicates the equivalence of mass/energy and information/entropy [8, 9]. From this frame, we can see that negative information/entropy, and hence negative mass/energy, does not describe particles and objects; instead, it describes the entanglement of particles and objects within a larger whole—i.e., within a quantum wavefunction [10]. From this perspective, it makes sense that we cannot objectively observe negative mass/energy. An experience of negative mass/energy implies a direct experience of the entanglement (the non-differentiability) of all things within our field of experience. Such a non-dual experience, equivalent to a dissolution of the subject-object division, is not an objective experience of discrete objects framed within a Euclidean spacetime governed by a chain of cause-and-effect, but rather, an integration of subject and object within a macroscopic complex projective space pointed to by this paradox in present physical theory.

 

The paradox of instantaneous virtual field propagation

It is well known that nothing with real mass/energy can travel faster than the speed of light. If anything with real mass/energy were to travel faster than the speed of light, it would violate causality—i.e., a coherent order of cause and effect. However, it may not be well known that changes in virtual fields do propagate instantaneously. This has been demonstrated in frustrated internal reflection experiments [4, 5]. Such changes in virtual fields do not constitute radiated light—they are fields associated locally with a physical object. Within the framework of quantum-electrodynamics, such fields are equivalent to the electrostatic and magnetostatic fields. We are familiar with instantaneous quantum phenomena over distance, such as quantum tunneling and quantum coupling. These instantaneous quantum phenomena are understood not to violate causality because the states on either side of the phenomenon (such as the states of two quantum-coupled particles) are mutually causal—i.e., they cannot be known independently. The paradox of instantaneous virtual field propagation is the implication of a mutually-causal relationship between macroscopic objects that are charged or magnetic (objects that possess electrostatic or magnetostatic fields) and something else, in spite of the fact that observable objects always appear to obey classical laws of cause and effect in relation to their environment.

Conceptual resolution of the paradox: If we were to identify a moment in which the substance of the particles and the objects they now compose were in mutually-causal relationship, it would be at the point of the Big Bang (and perhaps equivalently within black holes), where the curvature of time equals the curvature of space, such that space and time are non-differentiable. After this moment—or rather, as time appears to the observer as a Euclidean dimension orthogonal to space—the objects populating space appear disentangled from each other, acting on each other through a chain of cause-and-effect. ‘Re-entangling’ these objects would imply running time in reverse (in violation of the Law of Entropy) through an impossibly complex and intricately-coordinated series of events, or alternatively, by forcing them together into a black hole. This essay presents an argument that, in a certain sense, the objects we observe are already entangled within a macroscopic complex projective space that is within our capacity to experience at any moment as conscious entities. The implication of instantaneous virtual field propagation is that an objectively observable component of a macroscopic quantum wavefunction is present within Euclidean spacetime in the form of electrostatic and magnetostatic fields.

 

Bibliographical references

[1] Stillwell, John (2005). The Four Pillars of Geometry, Springer Press, ISBN 978-1-4419-2063-8

[2]  Ashtekar, Abhay and Schilling, Troy A. (1997). “Geometrical Formulation of Quantum Mechanics” arXiv:gr-qc/9706069v1

[3] Peskin, M.E., Schroeder, D.V. (1995). An Introduction to Quantum Field Theory, Westview Press, ISBN 0-201-50397-2

[4] Nimtz, G. (2011). “Tunneling Violates Special Relativity”.  Foundations of Science, 41: 1193-1199. arXiv:1003.3944  Note by B. Werner: The interpretation in this citation that tunneling violates special relativity is not correct, given the interpretation that the energy in the instantaneous field changes are purely imaginary. However, the data is nonetheless relevant to the argument presented in this essay.

[5] Eckle, P. et al.  (2008).  “Attosecond Ionization and Tunneling Delay Time Measurements in Helium”.  Science, 322: 1525.

[6] Gabriel Unger, Nikodem Poplawski. (2019) “Big bounce and closed universe from spin and torsion.” Astrophys. J. 870, 78. arXiv:1808.08327

[7] Planck Collaboration, (2015) “Planck 2015 results. XIII. Cosmological Parameters.” Astronomy and Astrophysics. arxiv.org/abs/1502.01589

[8]  Bekenstein, Jacob D. (1981). “Universal upper bound on the entropy-to-energy ratio for bounded systems”. Physical Review D (Particles and Fields), Volume 23, Issue 2, 15 January 1981, pp.287-298. 10.1103/PhysRevD.23.287

[9]  Shoichi Toyabe et al. (2010). “Information heat engine: converting information to energy by feedback control”. Nature Physics 6, 988-992. arXiv:1009.5287

[10]  N.J. Cerf and C. Adami. (1997). “Negative entropy and information in quantum mechanics”  arxiv.org/abs/quant-ph/9512022v3