In 2014 Steven Weinberg noted that quantum mechanics can avoid various difficulties, such as the many worlds hypothesis, by taking the quantum states to be density matrices without reference to state vectors. An immediate consequence of Weinberg's idea is that electron spin can be taken to follow SO(3) instead of SU(2). This radical departure from present understanding motivates our exploration of density matrices as a method of going beyond the Standard Model. An important tool for Standard Model calculations is the Feynman path integral formulation of quantum field theory. When the path integral is Wick rotated from time to imaginary time or temperature it becomes a method of cooling down density matrices. While this does not show that one goes beyond the Standard Model by Wick rotation, it does show a close relation between this method of cooling density matrices and quantum field theory of the Standard Model. We explore these ideas and exhibit toy models with particle content and symmetry similar to the Standard Model.
In 2014 Steven Weinberg noted that quantum mechanics can avoid various difficulties, such as the many worlds hypothesis, by taking the quantum states to be density matrices without reference to state vectors. An immediate consequence of Weinberg's idea is that electron spin can be taken to follow SO(3) instead of SU(2). This radical departure from present understanding motivates our exploration of density matrices as a method of going beyond the Standard Model. An important tool for Standard Model calculations is the Feynman path integral formulation of quantum field theory. When the path integral is Wick rotated from time to imaginary time or temperature it becomes a method of cooling down density matrices. While this does not show that one goes beyond the Standard Model by Wick rotation, it does show a close relation between this method of cooling density matrices and quantum field theory of the Standard Model. We explore these ideas and exhibit toy models with particle content and symmetry similar to the Standard Model.
In the standard formulation of quantum mechanics, observables are represented by operators in a Hilbert space and states by vectors in it. Quantum field theory too is formulated in terms of field operators. States in field theory include the vacuum state, one particle states, two particle states and so on. One does not measure operators or state vectors, however; measurements involve eigenvalues of operators or transition probabilities between states. It turns out that one can formulate quantum mechanics and field theory without using operators. Indeed, the probability amplitude for a transition from a position q1 at time t1 to a position q2 at time t2 can be expressed as a path integral ∫ dt L(q(t),q˙(t)), (5.1) where q(t) runs over all possible functions satisfying the conditions q(t1) = q1, q(t2) = q2 (5.2) and the integral over q(t) is an integral over all such paths of the exponential evaluated for each path. The measure of integration is not well defined a priori, but it is possible to give prescriptions that make sense. In classical mechanics, positions at two different times determine a path from the second order differential equation of motion, which yields a definite solution if two conditions are imposed. In contrast, quantum mechanics does not allow us to talk about trajectories. The path integral formulation manifestly refers to the myriad paths that may be taken by a particle in going from one point to another. It goes so far as to assign a probability amplitude to each path, and it is related to the classical action corresponding to that path. However, it is complex and it also involves a superposition, characteristic of quantum mechanics.
Carlo Rovelli's ``relational interpretation'' of quantum mechanics tells us that our understanding of quantum states is limited to their interactions with other quantum states. This implies that we have no understanding of the symmetry properties of a state vector except when considered in relation to at least one other state vector. The SU(3)xSU(2)xU(1) symmetry of the Standard Model is derived from their interactions and therefore cannot necessarily be applied to a single state vector. Steven Weinberg showed that mixed density matrices can have symmetries that are not available to state vectors. We explore the symmetry properties of finite symmetry groups for mixed density matrices. Just as mixed density matrices can define mixed states that require multiple state vectors, we find that the symmetries of mixed density matrices define ``mixed symmetries'' that are similar in structure to the symmetry of the Standard Model. We explore the point group symmetries and show one that gives the Standard Model symmetry. With this symmetry, we can generalize the Pauli spin matrices to a set that has irreducible representations matching the Standard Model plus a dark matter candidate.
Having introduced the birth of quantum physics at the end of the last chapter, we now formally explain some aspects of quantum mechanics. Instead of the usual material one can find in standard textbooks, we will start by motivating the standard “trick” of Wick rotation via the heat equation, and the Black–Scholes equation in finance, which gives us the Schrödinger equation. We then explore how Wick rotation in a path integral works, and how temperature of a field theory is essentially cyclic imaginary time. Later on we will discuss quantum entanglement and the Bell inequality, as well as how the cosmos provides an arena to close a certain loophole in the Bell test. We also discuss how quantum mechanics can explain various phenomena in astrophysics and cosmology, such as: how the Sun shines; how white dwarfs and neutron stars are supported by degenerative pressure; how neutrino oscillation works and what is a Majorana particle; and what is “quantum cosmology”.
Abstract The focus of Chapter 12 is the Feynman path integral formulations of quantum mechanics and quantum statistical mechanics. The connection between real-time quantum propagator and the density matrix is established via the Wick rotation, and a Trotter expansion is used to derive discrete path integral expressions for both. The continuous limit of the discrete path integral is derived to yield the functional integral formulation of the path integral. The construction of estimators and their averages for computing quantum observables is discussed. Next, the path integral is extended to incorporate Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, and a useful recursive formula for the partition function is derived that accelerates convergence by building up N-body partition functions from those of smaller systems. Quantum free energies are discussed, and algorithms for the evaluation of path integrals using molecular dynamics or Monte Carlo are presented. The chapter concludes with a discussion of higher-order factorizations and ring-polymer contraction methods to reduce the computational overhead of path-integral calculations
If mathematics is the queen of science, then physics might well be the king. It successes are obvious. However, as a science, physics may have failed in one regard, and that is to explain what its basic concepts – such as state vectors, wavefunctions, and transformation matrices – actually represent. When studying quantum mechanics, it is, effectively, hard to keep up the initial enthusiasm, and those who branch out to other fields – which is most of us – quickly end up going through the motions only: we regurgitate models and equations and know how to solve the standard problems, so as to pass the exam, but then forget about them as soon as possible. This paper explores a very intuitive sentiment about the issue: the wavefunction is a rather ‘flat’ mathematical object – it is two-dimensional, basically – so it can’t do the trick, perhaps. In contrast, Maxwell’s equations have real vectors in them, which is why a deeper or more intuitive understanding of electromagnetism comes relatively easily. Indeed, when everything is said and done, we are just human beings living in three-dimensional space, and that is why vector equations (or systems of vector equations), as a mathematical tool, make sense to us. This paper further explores this sentiment. It also offers a way out by, predictably, presenting yet another possible physical interpretation of the wavefunction. More importantly (for the reviewer of this paper, at least), this paper offers a sensible response to the mainstream view that three-dimensional physical interpretations of the wavefunction cannot make any sense because of the weird 720° symmetry of the wavefunction when describing spin-1/2 particles (fermions or – for all practical purposes – electrons). The author does so by analyzing (1) Dirac’s belt trick more in detail – and what it implies in terms of the interaction between the observer and the object – as well as (2) Feynman’s derivation of the transformation matrices for spin-1/2 two-state systems.
This book covers the entire span of quantum mechanics whose developments have taken place during the early part of the twentieth century up till the present day. We start with the Rutherford-Bohr model of the atom followed by Schrodinger's wave mechanics with its application to the solution of calculating the energy spectrum of a particle in a box, the harmonic oscillator and finally the hydrogen atom. Heisenberg's matrix mechanics and its duality with Schrodinger's wave mechanics, quantum mechanics in the interaction picture. Dirac's relativistic theory of the electron exhibiting the spin of the electron as a relativistic effect when it interacts with an external electromagnetic field. Feynman's path integral approach to non-relativistic quantum mechanics with is a marvellous intuitive interpretation as a sum over paths and how classical mechanics is obtained from its limit as Planck' constant tends to zero, methods for computing the spectra of the Dirac Hamiltonian in a radial potential, quantum field theory as developed by Feynman, Schwinger, Tomonaga and Dyson for describing the interaction between electrons, positrons, and photons via propagators using both the operator theoretic expansions and Feynman's path integral. We also introduce time independent and time dependent perturbation theory in quantum mechanics with applications to quantum gate design for quantum computers forming a major part of the research conducted by the author's research group, Quantum noise introduced into the Schrodinger and Dirac's equation based on the Hudson-Parthasarathy quantum stochastic calculus in Boson Fock space, scattering theory and wave operators with applications to quantum gate design, some aspects of second quantization like the interpretation of Boson Fock space in terms of harmonic oscillator algebras and the BCS theory of superconductivity, Wigner-Mackey-Frobenius theory of induced representations of a group with applications to Wigner's theory of particle classification, Dirac's equation in a gravitational field and Yang-Mills non-Abelian gauge theories with application to the construction of unified quantum field theories and finally, the more recent theory of super-symmetry which is a Boson-Fermion unification theory. We have discussed the statistics of Boson's, Fermions and Maxwell-Boltzmann based on entropy maximization. The book is written in problem-solution format and it would be of use to physicists and engineers interested respectively in developing unified field theories and in the design of quantum gates. Note: T&F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.