Gravity: Superspace Quantization

March 25, 2009, 8:38 pm
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Introduction

By the 1960's several new attacks were launched on the theoretical underpinnings of the quantum gravity program by Arnowitt, Deser, Misner and Wheeler. The basic idea was simple: If you really want to find a quantum theory for gravity, you had better first find out just how the basic ideas of quantum mechanics and general relativity will have to be enlarged to include each other.

In ordinary quantum field theory, each field is considered to be superimposed upon spacetime and is characterized by a particular quantum configuration. This configuration is indexed by a unique set of quantum numbers that represent the associated particle's spin, energy and angular momentum. This configuration  changes in a well-defined way as the system evolves subject to external influences such as absorbinga photon. Each state would be assigned a probability, and the evolution of the system would be dictated by a fundamental 'probability wave'equation. 

The electron is described as a wave  who's shape depends on its specific energy, angular momentum and spin.  All of the possible quantum configurations of the electron are elements of a new kind of space called Hilbert Space. This is an infinite-dimensional space, one dimension for each of the quantum states of the electron. As the electron's wave function changes, the electron moves from one configuration to another in Hilbert Space.

Is Hilbert Space real? No, not in the sense that 4-D spacetime seems to be. Hilbert Space  is merely a fictitious mathematical space which, like a bookkeeper's ledger, is used to keep track of all of the possible quantum states of a system. So how should we carry over these basic ideas of quantum mechanics into quantum gravity? What will take the place of a quantum state, a wave function, or even Hilbert space? And what properties of the gravitational field and spacetime would take the place of the discrete quantum numbers that were previously used to index all of the infinitude of possible quantum states of a particle? The answers provided by Canonical Quantum Theory are both surprising and profoundly unsettling if true.  

In ordinary physics you study a system defined at any one instant by its position coordinates ( X, Y and Z )and its momentum coordinates ( Px, Py and Pz). The system can be evolved forward in time by writing down an equation that shows how the position and momentum coordinates of the system change in time: an equation called the Hamiltonian. The position and momentum coordinates are said to be 'canonical variables' of the system because X and Px, Y and Py and Z and Pz are intimately related to one another in pairs.

In the ADM formulation of quantum gravity, instead of defining the coordinates ( X, Y, Z) and their conjugates (Px, Py and Pz) you define the metric gij and a new quantity which represents the conjugate to gij called Pij . This is a far more complex problem than in ordinary mechanics because instead of working with the spatial coordinates of a particle (X, Y, Z), you now work with the complete metric of space, gij and its derivitive Pij .  The equation that represented the Hamiltonian of the particle system is now an equation that describes how an entire 3-dimensional geometry for the universe changes from moment to moment. Time, in this sense, is just a new variable that indexes each of the 3-geometries of space.

Just as quantum mechanics required Hilbert Space to keep track of the possible configurations of a quantum system in space, in the ADM formulation, quantum gravity will also need its own analog to Hilbert Space in which every single possible state of the universe's 3-dimensional geometry is represented. This arena for quantum gravity is called 'Superspace'. As John Wheeler puts it, 

"Superspace is the arena of geometrodynsmics. The dynsmics of Einstein'scurved space geometry runs its course in Superspace as the dynamics of a particle unfolds in spacetime...The track of development of 3-geometry withtime [is] expressed as a sharp, thin 'leaf of history' that slices through superspace. The quantum principle replaces this deterministic account with a fuzzed-out leaf of history of finite thickness. In consequence, quantum fluctuations take place in the geometry of space that dominate the scene at distances of order the Planck length..."

This analysis also leads to some remarkable inferences. For example, the Heisenberg Uncertainty Principle denies the existence of a fixed classical spacetime. Spacetime only has a meaning in an average sense and at scales much larger than the Planck scale. Spacetime, and time itself, are not primary but are secondary ideas in nature and serve merely as approximations tosome perhaps more underlying reality. 

"...they have neither meaning nor application under circumstances where geometrodynamic effects become important...There is no spacetime, there isno time, there is no before, there is no after. The question of what happens 'next' is without meaning..."

Superspace, however, seemed to provide a key to the right way of looking at the classical problem of gravity, recovering general relativity as a by-product, and pointing the way towards how to quantize gravity from the geometric field perspective. Each of the 3-geometries in Superspace can be assigneda probability much as the quantum states of an electron in Hilbert Space. The evolution of space geometry is then seen in terms of a propagating wave of probability in Superspace which links the 3-geometry states together into a 4-D spacetime. The other almost incomprehensible aspect to Superspace is that each of the 3-geometries represent a snapshot of our entire spatial universe including the quantum states of ALL the fundamental fields, and their distribution in 3-dimensional space. This is an unimaginable level of complexity to deal with mathematically, and it is no wonder that theoreticians try to invoke various simplifying assumptions such as "only closed universes are considered with topologically compact manifolds", or just a single, often spinless,field is considered. These 'mini-Superspace' models are common, but do not do much justice to the complexity of working with the spatial configuration of the gravitational field. Topologically, even its 3-D, spatial shape can be far more complex. The topological view of spacetime predicts that as you approach the Planck scale at 10-33 centimeters, spacetime becomes awefully contorted, but at the same time you never seem to run out of spacetime coordinates themselves. The manifold developes wormhole 'handles' , but the surfaces of these handles consist of their own sub-planckian spacetimes patched together into an increasingly distorted spacetime.

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Gravitational waves are propagating gravitational fields, "ripples" in the curvature of space-time, generated by the motion of massive particles, such as two stars or two black holes orbiting each other. Gravitational waves cause a variable strain of space-time, which result in changes in the distance between points, with the size of the changes proportional to the distance between the points. Gravitational waves can be detected by devices which measure the induced length changes. Waves of different frequencies are caused by different motions of mass, and difference in the phases of the waves allow us to perceive the direction to the source and the shape of the matter that generated them.  (Source: NASA-The Laser Interferometer Space Antenna (LISA).)

Citation

Odenwald, Sten, Ph.D. (Contributing Author); Bernard Haisch (Topic Editor). 2008. "Gravity: Superspace Quantization." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published February 13, 2008].
<http://www.cosmosportal.org/articles/view/135656/>

 

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Citation

(2009). Gravity: Superspace Quantization. Retrieved from http://www.cosmosportal.org/view/article/135656

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