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R. Friedberg

Publications and source records attributed to R. Friedberg.

At least 19 recordsLinked to original sources
arXiv2017

What is quantum mechanics? A minimal formulation

This paper presents a minimal formulation of nonrelativistic quantum mechanics, by which is meant a formulation which describes the theory in a succinct, self-contained, clear, unambiguous and of course correct manner. The bulk of the presentation is the so-called \lq microscopic theory' (MIQM), applicable to any closed system $S$ of arbitrary size $N$, using concepts referring to $S$ alone, without resort to external apparatus or external agents. An example of a similar minimal microscopic theory is the standard formulation of classical mechanics, which serves as the template for a minimal quantum theory. The only substantive assumption required is the replacement of the classical Euclidean phase space by Hilbert space in the quantum case, with the attendant all-important phenomenon of quantum incompatibility. Two fundamental theorems of Hilbert space, the Kochen-Specker-Bell theorem and Gleason's theorem, then lead inevitably to the well-known Born probability rule. For both classical and quantum mechanics, questions of physical implementation and experimental verification of the predictions of the theories are the domain of the macroscopic theory, which is argued to be a special case or application of the more general microscopic theory.

physics.hist-ph↗
arXiv2010

Deviations of the Lepton Mapping Matrix from the Harrison-Perkins-Scott Form

We propose a simple set of hypotheses governing the deviations of the leptonic mapping matrix from the Harrison-Perkins-Scott (HPS) form. These deviations are supposed to arise entirely from a perturbation of the mass matrix in the charged lepton sector. The perturbing matrix is assumed to be purely imaginary (thus maximally $T$-violating) and to have a strength in energy scale no greater (but perhaps smaller) than the muon mass. As we shall show, it then follows that the absolute value of the mapping matrix elements pertaining to the tau lepton deviate by no more than $O((m_μ/m_τ)^2) \simeq 3.5 \times 10^{-3}$ from their HPS values. Assuming that $(m_μ/m_τ)^2 $ can be neglected, we derive two simple constraints on the four parameters $θ_{12}$, $θ_{23}$, $θ_{31}$, and $δ$ of the mapping matrix. These constraints are independent of the details of the imaginary $T$-violating perturbation of the charged lepton mass matrix. We also show that the $e$ and $μ$ parts of the mapping matrix have a definite form governed by two parameters $α$ and $β$; any deviation of order $m_μ/m_τ$ can be accommodated by adjusting these two parameters.

hep-ph↗
arXiv2009

A Timeon Model of Quark and Lepton Mass Matrices

It is proposed that $T$ violation in physics, as well as the masses of electron and $u, d$ quarks, arise from a pseudoscalar interaction with a new spin 0 field $τ(x)$, odd in $P$ and $T$, but even in $C$. This interaction contains a factor $iγ_5$ in the quark and lepton Dirac algebra, so that the full Hamiltonian is $P$, $T$ conserving; but by spontaneous symmetry breaking, the new field $τ(x)$ has a nonzero expectation value $<τ>\neq 0$ that breaks $P$ and $T$ symmetry. Oscillations of $τ(x)$ about its expectation value produce a new particle, the "timeon". The mass of timeon is expected to be high because of its flavor-changing properties. The main body of the paper is on the low energy phenomenology of the timeon model. As we shall show, for the quark system the model gives a compact 3-dimensional geometric picture consisting of two elliptic plates and one needle, which embodies the ten observables: six quark masses, three Eulerian angles $θ_{12}, θ_{23}, θ_{31}$ and the Jarlskog invariant of the CKM matrix. For leptons, we assume that the neutrinos do not have a direct timeon interaction; therefore, the lowest neutrino mass is zero. The timeon interaction with charged leptons yields the observed nonzero electron mass, analogous to the up and down quark masses. Furthermore, the timeon model for leptons contains two fewer theoretical parameters than observables. Thus, there are two testable relations between the three angles $θ_{12}, θ_{23}, θ_{31}$ and the Jarlskog invariant of the neutrino mapping matrix.

hep-ph↗
arXiv2008

Model with Strong $γ_4$ $T$-violation

We extend the $T$ violating model of the paper on "Hidden symmetry of the CKM and neutrino-mapping matrices" by assuming its $T$-violating phases $χ_\uparrow$ and $χ_\downarrow$ to be large and the same, with $χ=χ_\uparrow=χ_\downarrow$. In this case, the model has 9 real parameters: $α_\uparrow, β_\uparrow, ξ_\uparrow, η_\uparrow$ for the $\uparrow$-quark sector, $α_\downarrow, β_\downarrow, ξ_\downarrow, η_\downarrow$ for the $\downarrow$ sector and a common $χ$. We examine whether these nine parameters are compatible with ten observables: the six quark masses and the four real parameters that characterize the CKM matrix (i.e., the Jarlskog invariant ${\cal J}$ and three Eulerian angles). We find that this is possible only if the $T$violating phase $χ$ is large, between $-120^0$ to $-135^0$. In this strong $T$ violating model, the smallness of the Jarlskog invariant ${\cal J}\cong 3\times 10^{-5}$ is mainly accounted for by the large heavy quark masses, with $\frac{m_c}{m_t} <\frac{m_s}{m_b} \approx .02$, as well as the near complete overlap of $t$ and $b$ quark, with $(c|b)=-.04$.

hep-ph↗
arXiv2008

Theory of Timeon

It is proposed that $T$ violation in hadronic physics, as well as the masses of $u, d$ quarks, arise from a pseudoscalar interaction with a new spin 0 field $τ(x)$, odd in $P$ and $T$, but even in $C$. This interaction contains a factor $iγ_5$ in the quark Dirac algebra, so that the full Hamiltonian is $P$, $T$ conserving; but by spontaneous symmetry breaking, the new field $τ(x)$ has a nonzero expectation value $<τ>=τ_0$ that breaks $P$ and $T$ symmetry. Oscillations of $τ(x)$ about its expectation value $τ_0$ produce a new particle, the "timeon", whose mass is independent of any known quantities. If the timeon mass is within the range of present accelerators, observation of the particle can be helped with a search of $T$-violating events.}

hep-ph↗
arXiv2007

Jarlskog Invariant of the Neutrino Mapping Matrix

The Jarlskog Invariant $J_{ν-map}$ of the neutrino mapping matrix is calculated based on a phenomenological model which relates the smallness of light lepton masses $m_e$ and $m_1$ (of $ν_1$) with the smallness of $T$ violation. For small $T$ violating phase $χ_l$ in the lepton sector, $J_{ν-map}$ is proportional to $χ_l$, but $m_e$ and $m_1$ are proportional to $χ_l^2$. This leads to $ J_{ν-map} \cong {1/6}\sqrt{\frac{m_e}{m_μ}}+O \bigg(\sqrt{\frac{m_em_μ}{m_τ^2}}\bigg)+O \bigg(\sqrt{\frac{m_1m_2}{m_3^2}}\bigg)$. Assuming $\sqrt{\frac{m_1m_2}{m_3^2}}<<\sqrt{\frac{m_e}{m_μ}}$, we find $J_{ν-map}\cong 1.16\times 10^{-2}$, consistent with the present experimental data.

hep-ph↗
arXiv2007

Hidden Symmetry of the CKM and Neutrino Mapping Matrices

We propose that the smallness of the light quark masses is related to the smallness of the T violation in hadronic weak interactions. Accordingly, for each of the two quark sectors ("upper" and "lower") we construct a 3\times 3 mass matrix in a bases of unobserved quark states, such that the "upper"and "lower" basis states correspond exactly via the $W^\pm$ transitions in the weak interaction. In the zeroth approximation of our formulation, we assume T conservation by making all matrix elements real. In addition, we impose a "hidden symmetry" (invariance under simultaneous translations of all three basis quark states in each sector), which ensures a zero mass eigenstate in each sector. Next, we simultaneously break the hidden symmetry and T invariance by introducing a phase factor e^{iχ} in the interaction for each sector. The Jarlskog invariant J_{CKM}, as well as the light quark masses are evaluated in terms of the parameters of the model. We find a simple relation with J_{CKM}=(m_dm_s/m_b^2)^{1/2}Aλ^3\cos(χ/2), with A and λthe Wolfenstein parameters. Setting J_{CKM}=3.08 \times 10^{-5}, m_b=4.7GeV, m_s=95MeV, A=0.818 and λ=0.227, we find m_d\cos^2(χ/2) \simeq 2.4MeV, consistent with the accepted value m_d=3-7MeV. We make a parallel proposal for the lepton sectors. With the hidden symmetry and in the approximation of T invariance, both the masses of e and ν_1 are zero. The neutrino mapping matrix V_νis shown to be of the same Harrison-Scott form which is in agreement with experiments. We also examine the correction due to T violation, and evaluate the corresponding Jarlskog invariant {\cal J}_ν.

hep-ph↗
arXiv2006

Iterative Solutions for Low Lying Excited States of a Class of Schroedinger Equation

The convergent iterative procedure for solving the groundstate Schroedinger equation is extended to derive the excitation energy and the wave function of the low-lying excited states. The method is applied to the one-dimensional quartic potential problem. The results show that the iterative solution converges rapidly when the coupling $g$ is not too small.

quant-ph↗
arXiv2006

A Possible Relation between the Neutrino Mass Matrix and the Neutrino Mapping Matrix

we explore the consequences of assuming a simple 3-parameter form, first without T-violation, for the neutrino mass matrix M in the basis $ν_e, ν_μ, ν_τ$ with a new symmetry. This matrix determines the three neutrino masses m_1, m_2, m_3, as well as the mapping matrix U that diagonalizes M. Since U, without T-violation, yields three measurable parameters $s_{12}, s_{23}, s_{13}$, our form expresses six measurable quantities in terms of three parameters, with results in agreement with the experimental data. More precise measurements can give stringent tests of the model as well as determining the values of its three parameters. An extension incorporating T-violation is also discussed.

hep-ph↗
arXiv2004

New Ways to Solve the Schroedinger Equation

We discuss a new approach to solve the low lying states of the Schroedinger equation. For a fairly large class of problems, this new approach leads to convergent iterative solutions, in contrast to perturbative series expansions. These convergent solutions include the long standing difficult problem of a quartic potential with either symmetric or asymmetric minima.

quant-ph↗
arXiv2001

A Convergent Iterative Solution of the Quantum Double-well Potential

We present a new convergent iterative solution for the two lowest quantum wave functions $ψ_{ev}$ and $ψ_{od}$ of the Hamiltonian with a quartic double well potential $V$ in one dimension. By starting from a trial function, which is by itself the exact lowest even or odd eigenstate of a different Hamiltonian with a modified potential $V+δV$, we construct the Green's function for the modified potential. The true wave functions, $ψ_{ev}$ or $ψ_{od}$, then satisfies a linear inhomogeneous integral equation, in which the inhomogeneous term is the trial function, and the kernel is the product of the Green's function times the sum of $δV$, the potential difference, and the corresponding energy shift. By iterating this equation we obtain successive approximations to the true wave function; furthermore, the approximate energy shift is also adjusted at each iteration so that the approximate wave function is well behaved everywhere. We are able to prove that this iterative procedure converges for both the energy and the wave function at all $x$.

quant-ph↗
arXiv2000

A New Method to Derive Low-Lying N-dimensional Quantum Wave Functions by Quadratures Along a Single Trajectory

We present a new method to derive low-lying N-dimensional quantum wave functions by quadrature along a single trajectory. The N-dimensional Schroedinger equation is cast into a series of readily integrable first order ordinary differential equations. Our approach resembles the familiar W.K.B. approximation in one dimension, but is designed to explore the classically forbidden region and has a much wider applicability than W.K.B.. The method also provides a perturbation series expansion and the Green's functions of the wave equation in N-dimension, all by quadratures along a single trajectory. A number of examples are given for illustration, including a simple algorithm to evaluate the Stark effect in closed form to any finite order of the electric field.

quant-ph↗
arXiv1999

Relations Between Low-lying Quantum Wave Functions and Solutions of the Hamilton-Jacobi Equation

We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\geq 0$, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller quantum tunneling parameter $ε$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $ε$, we show how the coefficients of $g^{-m}ε^n$ in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.

quant-ph↗
arXiv1997

Comment on "Boson-fermion model beyond the mean-field approximation"

In a recent paper [A.S.Alexandrov, J.Phys.:Condens.Matter 8, 6923 (1996); cond-mat/9603111], it has been suggested that there is no Cooper pairing in boson-fermion models of superconductivity. We show that this conjecture is based on an inconsistent approximation that violates an exact identity. Quite generally, the divergence of the fermion t-matrix (the Thouless criterion) is accompanied by the condensation of a boson mode.

cond-mat.supr-con↗