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Path Integral representation for Polymer Quantized Scalar Fields

According to loop quantum gravity, matter fields must be quantized in a background independent manner. For scalar fields, such a background independent quantization is called polymer quantization and is inequivalent to the standard Schrodinger quantization. It is therefore important to obtain predictions from the polymer quantized scalar field theory and compare with the standard results. As a step towards this, we develop a path integral representation for the polymer quantized scalar field. We notice several crucial differences from the path integral for the schrodinger quantized scalar field. One important difference is the appearance of an extra summation at each point in the path integral for the polymer quantized theory. A second crucial difference is the loss of manifest Lorentz symmetry for a polymer quantized theory on Minkowski Space.