Quantum Mechanics (Book): Difference between revisions

A particular sub-Lie algebra is determined as a representation. For wavefunctions on <math> \mathbb{R}^3 </math>, there are six distinguished operators: position operators are multiplication by each of the three coordinate functions (e.g. <math> x * \Psi (x,y,z) </math> , momentum operators are i times the partial derivatives with respect to the coordinates (e.g. <math> i \frac{\partial}{\partial x}\Psi (x,y,z)</math>). Writing out the six-by-six table of commutators gives us a Lie algebra, with a particular structure known as the (three dimensional) Heisenberg Lie algebra. Note the similarity to the Poisson brackets of position and momentum coordinate functions in classical mechanics. In infinite dimensions, Hilbert spaces cannot be as easily distinguished as in finite dimensions by the dimension, but finding a representation of the Heisenberg Lie algebra fixes the dimension of the coordinate space on which quantum mechanics happens. The uniqueness of this representation was established by Stone and Von Neumann, coined and generalized by Mackey. Basic representations in quantum mechanics are explained in an example-focused approach in Woit's book, from classical Hamiltonian mechanics up to relativistic quantum mechanics and basic quantum field theory. There is also analytical content to the Heisenberg representation - note that the momentum operators have plane-waves as eigenvectors which are not normalizable, and an eigenvector of the position operator would have to be nonzero at a single position yet still have an integral of one, a property no function has. Allowing for a continuous spectrum, we must also consider vectors/functions outside of the Hilbert space, leading to the concept of generalized functions and Rigged Hilbert spaces worked in detail by Gel'fand and Vilenkin in volume 4 of Generalized Functions (an original series on the subject). The continuous analog of matrices and their spectrum are fully explored here via the Schwartz kernel theorem and Spectral theorem, with discussion of the properties of the Heisenberg representation. Finite dimensional Hilbert spaces occur physically quite late in Landau's book, as they do not pertain to probabilities over spatial or momentum coordinates, but allow for the possibility of a discrete 'spin' argument of the wavefunction <math> \Psi (x,y,z,\sigma )</math>. These descend from considering possible Hilbert spaces obeying the laws of quantum mechanics also carrying a representation of the aforementioned Lie algebra of infinitesimal 3-d (or higher dimensional) rotations. In the Euclidean group, rotations intertwine with translations (which are generated by exponentiating the Lie algebra of partial derivatives) in a group-theoretic construction called the semidirect product. This also occurs in space-time with the Poincare group, where translations now also occur in time generated by the Hamiltonian operator, and in addition to rotations there are also Lorentz boosts. Representations utilizing the extra structure of the semidirect product are known as induced, and this is the magic that fixes the particle spins we see in quantum mechanics and (relativistic) quantum field theory. While Woit runs through the basic arguments, Mackey's book includes other examples and shows the full power of the technique in determining most of the properties of quantum mechanics deemed mysterious through this mathematical principle alone. Also note Gel'fand's book on the [https://www.google.com/books/edition/Representations_of_the_Rotation_and_Lore/_QZRDwAAQBAJ?hl=en&gbpv=0 Representations of the Rotation and Lorentz Groups] for an alternate presentation. These books mostly focus on lower-dimensional Lie groups, the higher dimensional theory and of more general semisimple Lie groups is covered by Fulton & Harris.

=== Applications ===