Reinterpreting Landauer conductance, solving the quantum measurement problem, grand unification

TL;DR

The paper argues that a local time and a local partial density of states ($\rho_{lpd}$), derived from a physical clock, can remain well-defined in open mesoscopic systems and may be negative in quantum regimes. It then reinterprets the Landauer-Buttiker conductance through this local-state framework, deriving zero-temperature ($G_{two-probe}^{zero-temperature}=\frac{e_0^2}{h}|t(E)|^2$) and finite-temperature ($G_{two-probe}^{finite-temperature}=\frac{e_0}{h} \int dE\,(-\frac{\partial f}{\partial E})|t(E)|^2$) results and clarifying the role of lead DOS ($\rho_{pd}=1/(hv)$) in transport. A three-probe STM setup is used to show that the measured coherent current change relates to $\rho_{lpd}$ via $|s'_{\alpha\gamma}|^2 - |s_{\alpha\gamma}|^2 = -2\pi\,\rho_{lpd}(E,\alpha,\mathbf{r},\gamma)$, validating $\rho_{lpd}$ even when negative (e.g., at Fano resonances) and connecting measurement to a deterministic local state through topological Argand-diagram considerations. The authors claim this framework provides a deterministic account of quantum measurement and a relativistically consistent notion of time, supporting a grand unification of classical and quantum laws at low energies without invoking quantum gravity.

Abstract

In a series of recent papers we have proved rigorously that time travel is a reality and very much feasible by using quantum mechanical processes. There are plenty of indirect experimental support untill a direct experiment is conducted. The process crucially depend on the reality of a local time as well as a local partial density of states (LPDOS) that can become negative very easily in the quantum regime of mesoscopic systems. Mesoscopic systems are small enough to allow us to experimentally access the intermediate regime between the classical and quantum worlds. This LPDOS is in every sense a hidden variable in quantum mechanics that does not show up in the axiomatic framework of quantum mechanics. It can be inferred through physical clocks obeying quantum dynamics and can be rigorously justified from the properties of the Hilbert space that is uniquely isomorphic to the complex plane. Therefore one can naturally guess that LPDOS will have something important to say about quantum measurement as well as the unification of classical and quantum laws. We therefore undertake the exercise to show that LPDOS can very much allow us to re-interpret the enormously successful phenomenological Landauer-Buttiker formalism for mesoscopic systems and put it on firm theoretical ground as a bridge between classical and quantum mechanics, thereby unifying them. Essentially the local time calculated quantum mechanically can dilate exactly like the proper time of relativity and be consistent with the coordinate time of relativity. Also the measured conductance of mesoscopic samples is a deterministic quantum measurement outcome from a linear superposition of states, essentially because of LPDOS, which solves the quantum measurement problem. For this we analyze the three probe conductance formula in details and give our arguments for the general case.

Figures (10)

• Figure 1: Fig. 1. A 1D quantum scatterer is sandwiched between two classical reservoirs. The source reservoir is to the left and the sink reservoir is to the right of the two wavy lines in the figure. The scatterer in between is shown in the form of a finite square barrier, whose transmission amplitude is discussed in almost all text books of quantum mechanics. The horizontal axis is the 1D coordinate axis (say $x$-axis) and the vertical axis is the energy axis. The origin of the energy axis is the band bottom of the conduction band making the sample and the height of the square barrier is shown by an arrow. Such a 1D square barrier with leads can be practically made from the GaAs-AlGaAs interface by creating a narrow confinement potential in the lateral $y$ and $z$ directions. Since the reservoirs are classical, they have a statistically large number of electrons and the system in between has only quantum states that can transmit electrons from the left reservoir to the right. The two reservoirs can have different chemical potentials and different temperatures that can be seen from the Fermi distributions generally shown inside the two reservoirs.
• Figure 2: Fig. 2. The basic set up for a three probe Landauer conductance where there are only two fixed leads indexed $\gamma$ and $\alpha$ apart from the STM tip $\beta$.
• Figure 3: Fig. 3. The sample is the three prong potential shown by the solid lines and the entire system consist of the sample connected to three reservoirs via three leads. Different regions of the system is marked by Roman numbers, I, IV and VII being the leads, shown by dashed lines. Lead 2 is similar to the lead $\beta$ in Fig. 2 but earthed and the chemical potential of lead 3 is also set to zero. In that case, conventionally, a small positive chemical potential at lead 1 will inject a current while the two other leads carry current away from the sample. Lead 2 connects to the sample through a tunneling barrier shown as region VI. Lengths of different regions is shown as $l_2$, $l_3$ etc. This system is a 1D version of the system in Fig. 2 where lead $\gamma$ is renamed as lead 1, etc, and as a result $t_{31}\equiv s_{31}$, $t_{21}\equiv s_{21}$, and $r_{11}\equiv s_{11}$.
• Figure 4: Fig. 4. In this figure we plot the AD for $t_{31}$ of the system shown in Fig. 3 for the coupling potential $U_1$ varying in a range that give three sub-loops, all within one Riemann surface. The starting point is marked by a small square block corresponds to $U_1l=-10000$. The end point is unmarked and corresponds to a value $U_1l=-10$. All the sub-loops smoothly come back to a point marked by a cross. Other parameters are mentioned inside the figure.
• Figure 5: Fig. 5. In this figure we are plotting the LHS and RHS of Eq. 65 to show that they both can oscillate identically between positive and negative values. The primed and unprimed values are for small differences in $U_1$ at a $k$ value that we vary continuously. The sign change and magnitude of both the curves originate from the smooth cyclic AD of Fig. 4.