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A Modern Approach to Born Reciprocity

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Morgan, SO (2010) A Modern Approach to Born Reciprocity. PhD thesis, University of Tasmania.

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Abstract

In the early twentieth century, Max Born attempted to develop a theory he called
the “principle of reciprocity”. He observed that many formulae of physics remain
unchanged under the following transformation of space-time coordinates and
momentum-energy variables: xμ -> pμ; pμ -> −xμ. Examples include Hamilton's
equations and Heisenberg's commutation relations. Born's attempts to expand
this observation to a general theory were largely unsuccessful. More recently,
Stephen Low has made use of the group theoretical methods of Eugene Wigner,
Valentine Bargmann and George Mackey to study a group which possesses Born
reciprocity as an intrinsic symmetry, called the "quaplectic group", Q(1, 3). This
involves the postulation of a new fundamental constant: the maximum rate of
change of momentum (or maximum force) - denoted by b. It also involves a new
space-time-momentum-energy line element:
ds2 = −dt2 +
1
c2
dx2 +
1
b2 (dp2 −
1
c2
de2),
which remains invariant under quaplectic transformations.
In this work we consider the different contraction limits of the quaplectic group
in analogy to the contraction of the Poincar´e group to the Galilei group in the limit
c -> 1. In particular, the quaplectic group contracts to the Poincar´e group in the
limit b -> 1 (under the constraint that the reference frames must be inertial) and
to the Hamilton group - the group of non-inertial classical mechanics - in the limit
b, c -> 1. For the compact group Q(2) acting on two dimensional Euclidean space
we consider the branching rules and use the P2 Casimir operator of the Euclidean
subgroup to label states. We consider the implications of Born reciprocity to
the Schr¨odinger-Robertson inequality, concluding that the covariance matrix
is quaplectic invariant and that physically distinct semi-classical limits of two
different but unitarily-equivalent minimal uncertainty states must be related by a
unitary transformation which does not belong to the quaplectic group. Finally, we
explore the worldline quantisation of a system invariant under reciprocal relativity,
finding that the square of the energy-momentum tensor is continuous over the
entire real line. The resulting states therefore include tachyonic and null states as
well as massless states of continuous spin which cannot be projected out in the
current formulation. These states are discussed along with the massive states.

Item Type: Thesis (PhD)
Keywords: reciprocity, relativity, group representations, high energy physics, Max Born
Date Deposited: 04 Apr 2011 01:43
Last Modified: 11 Mar 2016 05:53
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