Generalized uncertainty principles and quantum field theory

Quantum mechanics with a generalized uncertainty principle arises through a representation of the commutator $[\widehat{x},\widehat{p}]=if(\widehat{p})$. We apply this deformed quantization to free scalar field theory for $f_{\pm }=1\pm \beta p^2$. The resulting quantum field theories have a rich fine scale structure. For small wavelength modes, the Green’s function for $f_{+}$ exhibits a remarkable transition from Lorentz to Galilean invariance, whereas for $f_{-}$ such modes effectively do not propagate. For both cases Lorentz invariance is recovered at long wavelengths.

Figure 1

Potentials appearing in the eigenvalue equation (7b) for various choices of $f$. We take $g=0.225$. Note that the number of bound states in the last case is directly determined by the height of the finite potential; i.e., it is a function of $g=k/M_{\star }$.

Figure 2

Matrix elements $M_{\star }|c_n{|}^2$ for $f=1+x^2$ (red/light gray), $f=1-x^2$ (blue/dark gray), and $f=1$ (purple dashed). These are proportional to the residues associated with the resonant poles of the Green’s function.

Figure 3

Bound state contributions to the Green’s function in the complex $\omega$ plane for $f=1\pm x^2$. Each pair of spikes indicates a resonant mode. The $f=1-x^2$ case has only two bound states for this choice of parameters, and hence exhibits fewer poles than the $f=1+x^2$ case.

Figure 4

Dispersion relations for the principal $n=1$ resonance of $G(\omega ,\mathbf{k})$ for various choices of $f$. Note that for the $f=1-x^2$ case, the $n=1$ bound state transitions to the continuum at $k=M_{\star }/\sqrt{2}$; hence the termination of the $\omega =\omega (k)$ curve. Dispersion relations for the higher order excitations are qualitatively similar.

Figure 5

Scattering to bound state matrix elements $|c(\nu ){|}^2$ for the $f(x)=1-x^2$ case.