The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative nonrelativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.

It is known that Berry curvature of the band structure of certain crystals can lead to effective noncommutativity between spatial coordinates. Using the techniques of twisted quantum field theory, we investigate the question of the formation of a paired state of twisted fermions in such a system. We find that to leading order in the noncommutativity parameter, the gap between the non-interacting ground state and the paired state is smaller compared to its commutative counterpart. This suggests that BCS type superconductivity, if present in such systems, is more fragile and easier to disrupt.

We derive the thermal correlators for twisted quantum fields on noncommutative spacetime. We show that the thermal expectation value of the number operator is same as in commutative spacetime, but that higher correlators are sensitive to the noncommutativity parameters θ µν .General arguments involving classical gravity and quantum uncertainties suggest that spacetime structure should be "granular" at very short distances [1]. A specific model for this granularity is realized by the Groenewold-Moyal (GM) plane, where instead of the usual pointwise product (f · g)(x) on R d+1 , one works with the noncommutative product (f * g)(x) = f (x)e

In the present work, we study the properties of dissipative shocks for fully relativistic accretion flows around spinning black holes. In an accretion flow harboring a dissipative shock (formally known as radiative shock), a significant portion of the thermal energy may get released from the post-shock corona. A stellar-mass black hole may therefore emit hard X-rays from the inner edge of the disk. If the bulk energy loss is significant, post-shock pressure drops, and shock moves forward towards the black hole compressing the size of the post-shock corona, resulting an enhancement of the corona temperature and compression ratio. The dynamical properties of the radiative shocks are therefore systematically investigated to understand accurately the radiative loss processes, temporal variations, and the spectral properties. We notice that the range of flow parameters (e.g. energy and angular momentum) responsible for the formation of ‘shocks in accretion (SA)’ are identical for both the case of standing and dissipative shocks. The spin of the black hole enhances the dissipation further. We estimate the maximum energy release, which is observed close to $100\%$ in the extreme cases. This could be useful in explaining various observed phenomena namely the formation and the systematic evolution of QPOs, and the time lags in between hard and soft X-ray photons (e.g., XTE J1550-564 and GRO J1655-40 etc.) during their outbursts.

Following the original approach of Maxwell-Boltzmann(MB), we derive a 4-velocity distribution function for the relativistic ideal gas. This distribution function perfectly reduces to original MB distribution in the nonrelativistic limit. We express the relativistic equation of state(EOS), ρ−ρ 0 = (γ −1) −1 p, in the two equations: ρ = ρ 0 f (λ), and p = ρ 0 g(λ), where λ is a parameter related to the kinetic energy, hence the temperature, of the gas. In the both extreme limits, they give correct EOS: ρ = 3p in the ultra-relativistic, and ρ − ρ 0 = 3 2 p in the non-relativistic regime. Using these equations the adiabatic index γ (= cp cv ) and the sound speed a s are calculated as a function of λ. They also satisfy the inequalities: 4 3 ≤ γ ≤ 5 3 and a s ≤ 1 √ 3 perfectly.