Nonperturbative Quantum Field Theory and the Structure of Matter

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As far as I see, nothing in loop quantum gravity suggests that one could compute masses from quantum gravity. Origin of the Universe. It is likely that a sound quantum theory of gravity will be needed to understand the physics of the Big Bang. The converse is probably not true: We should be able to understand the small scale structure of spacetime even if we do not understand the origin of the Universe.

Perturbative and Nonperturbative Aspects of Quantum Field Theory

Arrow of time. Roger Penrose has argued for some time that it should be possible to trace the time asymmetry in the observable Universe to quantum gravity. Physics of the mind. Penrose has also speculated that quantum gravity is responsible for the wave function collapse, and, indirectly, governs the physics of the mind [ ]. A problem that has been repeatedly tied to quantum gravity, and which loop quantum gravity might be able to address, is the problem of the ultraviolet infinities in quantum field theory. The very peculiar nonperturbative short scale structure of loop quantum gravity introduces a physical cutoff.

Since physical spacetime itself comes in quanta in the theory, there is literally no space in the theory for the very high momentum integrations that originate from the ultraviolet divergences. Lacking a complete and detailed calculation scheme, however, one cannot yet claim with confidence that the divergences, chased from the door, will not reenter from the window. Here, I begin the technical description of the basics of loop quantum gravity. The starting point of the construction of the quantum theory is classical general relativity, formulated in terms of the Sen-Ashtekar-Barbero connection [ , 8 , 41 ].

Detailed introductions to the complex Ashtekar formalism can be found in the book [ 10 ], in the review article [ ], and in the conference proceedings [ 81 ]. The real version of the theory is presently the most widely used. Classical general relativity can be formulated in phase space form as follows [ 10 , 41 ].

We use a,b , The internal indices can be viewed as labeling a basis in the Lie algebra of SU 2 or in the three axis of a local triad.


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We indicate coordinates on M with x. The theory is invariant under local SU 2 gauge transformations, three-dimensional diffeomorphisms of the manifold on which the fields are defined, as well as under coordinate time translations generated by the hamiltonian constraint. The full dynamical content of general relativity is captured by the three constraints that generate these gauge invariances [ , 10 ]. As already mentioned, the Lorentzian hamiltonian constraint does not have a simple polynomial form if we use the real connection 2.

For a while, this fact was considered an obstacle to defining the quantum hamiltonian constraint; therefore the complex version of the connection was mostly used. However, Thiemann has recently succeeded in constructing a Lorentzian quantum hamiltonian constraint [ , , ] in spite of the non-polynomiality of the classical expression. This is the reason why the real connection is now widely used. Certain classical quantities play a very important role in the quantum theory. See [ 77 ] for more details. These are the loop observables, introduced in Yang Mills theories in [ 95 , 96 ], and in gravity in [ , ].

The loop observables coordinatize the phase space and have a closed Poisson algebra, denoted by the loop algebra. This algebra has a remarkable geometrical flavor.

Nonperturbative Quantum Field Theory and the Structure of Matter - INSPIRE-HEP

More precisely. A non-SU 2 gauge invariant quantity that plays a role in certain aspects of the theory, particularly in the regularization of certain operators, is obtained by integrating the E field over a two dimensional surface S.

The physical interpretation of the theory is based on the connection between these operators and classical variables, and on the interpretation of as the space of the quantum states. The dynamics is governed by a hamiltonian, or, as in general relativity, by a set of quantum constraints, constructed in terms of the elementary operators. To assure that the quantum Heisenberg equations have the correct classical limit, the algebra of the elementary operator has to be isomorphic to the Poisson algebra of the elementary observables.

In other words, define the quantum theory as a linear representation of the Poisson algebra formed by the elementary observables. For the reasons illustrated in section 5 , the algebra of elementary observables we choose for the quantization is the loop algebra, defined in section 6. Thus, the kinematic of the quantum theory is defined by a unitary representation of the loop algebra. Here, I construct such representation following a simple path.

These functionals form a linear space, which we promote to a Hilbert space by defining a inner product. For technical reasons, we require the links to be analytic.


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U i A is an element of SU 2. Pick a function f g These states are called cylindrical states because they were introduced in [ 15 , 16 , 17 ] as cylindrical functions for the definition of a cylindrical measure. Thus, given any two cylindrical functions, we can always view them as having the same graph formed by the union of the two graphs.

Loop Quantum Gravity

Given this observation, we define the scalar product between any two cylindrical functions [ , 15 , 16 , 17 ] by. This scalar product extends by linearity to finite linear combinations of cylindrical functions. It is not difficult to show that 14 defines a well defined scalar product on the space of these linear combinations. Completing the space of these linear combinations in the Hilbert norm, we obtain a Hilbert space. This is the unconstrained quantum state space of loop gravity.

An important property of the scalar product 14 is that it is invariant under both these transformations. At first sight, this may seem a serious obstacle to its physical interpretation. But we will see below that, after factoring away diffeomorphism invariance, we may obtain a separable Hilbert space see section 6. Also, standard spectral theory holds on , and it turns out that using spin networks discussed below one can express as a direct sum over finite dimensional subspaces which have the structure of Hilbert spaces of spin systems; this makes practical calculations very manageable.

Finally, we will use a Dirac notation and write. A subspace of is formed by states invariant under SU 2 gauge transformations.

General Information

We now define an orthonormal basis in. This basis represents a very important tool for using the theory. It was introduced in [ ] and developed in [ 34 , 35 ]; it is denoted spin network basis. These sates are called loop states. Using Dirac notation, we can write. It is easy to show that loop states are normalizable. Products of loop states are normalizable as well. Multi- loop states represented the main tool for loop quantum gravity before the discovery of the spin network basis.

Linear combinations of multiloop states over- span , and therefore a generic state g A is fully characterized by its projections on the multiloop states, namely by. Equation 19 can be explicitly written as an integral transform, as we will see in section 6. Associate an invariant tensor v in the tensor product of the representations s An invariant tensor is an object with n indices in the representations s An invariant tensor is also called an intertwining tensor.

All invariant tensors are given by the standard Clebsch-Gordon theory. More precisely, for fixed s Pick a basis v j is this space, and associate one of these basis elements to the node.

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Notice that invariant tensors exist only if the tensor product of the representations s This yields a condition on the coloring of the links. It was Penrose who first had the intuition that this mathematics could be relevant for describing the quantum properties of the geometry, and who gave the first version of spin network theory [ , ]. We take the propagator of the connection along each link of the graph, in the representation associated to that link, and then, at each node, we contract the matrices of the representation with the invariant tensor. The spin network states are normalizable.

The normalization factor is computed in [ 77 ].