Abstract
We propose a new inner product for scalar fields that are solutions of the Klein-Gordon equation with $m^2<0$. This inner product is non-local, bearing an integral kernel including Bessel functions of the second kind, and the associated norm proves to be positive definite in the subspace of oscillatory solutions, as opposed to the conventional one. Poincar\'e transformations are unitarily implemented on this subspace, which is the support of a unitary and irreducible representation of the proper orthochronous Poincar\'e group. We also provide a new Fourier Transform between configuration and momentum spaces which is unitary, and recover the projection onto the representation space. This new scenario suggests a revision of the corresponding quantum field theory.
Highlights
The Klein-Gordon equation with a negative squared mass, m2 1⁄4 −κ2 < 0, generated an interesting debate in the late 1960’s and the 1970’s
Regardless of the nature of tachyonic modes, either fundamental or not, or their actual role in any physical theory, our aim in this paper is to point out that a description of the scalar solutions of (1) consistent with a unitary and irreducible representation of the Poincaregroup cannot rely on the standard inner product, Z hφ; φist 1⁄4 i dσμðφðxÞÃ∂μφðxÞ − ∂μφðxÞÃφðxÞÞ; ð2Þ
The inner product (4) turns out to define a positive definite norm on the space of oscillatory solutions, which is key to establish the equivalence between the configuration space representation and the known momentum space representation, with support on the one-sheet hyperboloid and which bears a positive definite inner product: the well-known expression (2) needs to be discarded for the configuration space representation as it fails to be positive definite
Summary
The Klein-Gordon equation with a negative squared mass, m2 1⁄4 −κ2 < 0, generated an interesting debate in the late 1960’s and the 1970’s. The theory did not succeed; one of the main issues being the problems with the causality [2] and the unitary implementation of Poincaretransformations on this simple model [3,4,5,6,7] In this sense, the representations of the Poincaregroup for m2 < 0 were not originally analyzed by Wigner [8], later authors constructed. Regardless of the nature of tachyonic modes, either fundamental or not, or their actual role in any physical theory, our aim in this paper is to point out that a description of the scalar solutions of (1) consistent with a unitary and irreducible representation of the Poincaregroup cannot rely on the standard inner product,. Poincareinvariance fixes the form and coefficients of the kernels k1 and k2 and their covariant version K
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