Editing Quantum Mechanics (Book) (section)

{{InfoboxBook
|title=Quantum Mechanics
|image=[[File:Landau Quantum Mechanics.jpg]]
|author=[https://en.wikipedia.org/wiki/Lev_Landau Lev Landau]
|language=English
|series=Course of Theoretical Physics
|genre=
|publisher=Butterworth Heinemann
|publicationdate=1977
|pages=677
|isbn13=978-0-7506-3539-8
}}

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Quantum Mechanics is mathematically simple compared to the previous two foundations of physics - classical mechanics and fields. It is linear, so no complicated manifolds like in mechanics or relativity. This is deceptive, but is a helpful crutch when first learning the subject. The true nature of quantum mechanics is geometric: the projective geometry of Hilbert space, geometric quantization from classical/symplectic geometry to quantum phase spaces, the moment map in symplectic geometry gives the map to the convex space of probability distributions, the bundles and connections appearing in the quantum Hall effect, and finally it is also a task to precisely interpret generalized functions, spectra of self-adjoint operators on function spaces, (projective) unitary group representations, and kernels/matrices of these general operators.

After accumulating enough experimental evidence, mathematical methods began to form for Quantum Mechanics when Heisenberg re-invented matrices and matrix multiplication: utilizing infinite matrices to keep track of probabilities of measurements, transition frequencies, successive measurements. A more natural starting point was realized later, by Schrodinger and fully expanded upon by Von Neumann, to encode the state as a complex-valued function Ψ(x,y,z) e.g. of the spatial coordinates. The bases determining the components of Heisenberg's infinite matrices then became countable collections of such functions, and in this function-state-space framework, the matrices were upgraded to operators that could also have a continuous rather than just a discrete/countable spectrum. So far, this is just the extension of complex linear algebra to infinite dimensions. However, there is the added restriction on the observable quantity's operators for their spectrum to be real. This comes from self-adjointness, which allows us to easily choose bases in which the operator is diagonalized, and the algebraic property that operator commutators between them remain self-adjoint. Now, the objects have physical interpretations:
* Hermitian norm-one vectors <math> \rightarrow </math> states
* self-adjoint operators <math> \rightarrow </math> observable quantities
* eigenvalues of self-adjoint operators <math> \rightarrow </math> possible measured values of the observable
* eigenvectors of self-adjoint operators <math> \rightarrow </math> states with a definite value of the observable
* Hermitian inner-product of a general vector with an eigenvector, components in an eigenbasis <math> \rightarrow </math> the real magnitude resulting complex value gives the probability/weight of the state's value of the observable being the associated eigenvalue
These details are recalled compactly in the first two chapters of Landau's book. Restating them here, we give connections to the underlying mathematics. Choosing a particular observable <math> H </math> to be the energy determines the time evolution of the system, just as in classical Hamiltonian mechanics. In quantum mechanics, this is realized as Schrodinger's equation, but mathematically it is just the property that its operator exponential <math> e^{-itH} </math> is unitary thus preserving the inner-product/unit sphere of the function/Hilbert space. This is where the second main mathematical ingredient of quantum mechanics appears - Lie group and Lie algebra representations. Stone's theorem (on one-parameter groups) tells us that the embedding of the real line into the space of operators that e.g. picks the family of unitary operators determining time evolution only needs continuity properties to be differentiable. This leads us to turn from continuous unitary transformations to their simpler generating self-adjoint operators. The essence of Lie algebra representations is when one has multiple simultaneous families with nontrivial commutation relations. This sounds abstract, but appears as easily as in real linear algebra when transitioning from studying rotations in two to three dimensions. Differentiating the families of rotations around the principal axes gives three matrices whose commutators can be expressed in terms of one another - the structure of a Lie algebra. This is the same property that the collection of all self-adjoint operators have, in a given Hilbert space. 

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.

Why complex analysis? The Heisenberg Lie-algebra representation suggest wave functions can be equivalently chosen as functions of position or momentum, with the Fourier-transform relating the two descriptions of the wavefunction. Woit also shows that there is a basis in between, the Bargmann-Fock representation, that converts the situation into that of complex-valued functions of complex coordinates. Additionally, in proving the spectral theorem (e.g. via resolvents in [https://www.youtube.com/playlist?list=PLPH7f_7ZlzxQVx5jRjbfRGEzWY_upS5K6 Schuller's videos on Quantum Theory]) one finds that contour-integral ideas are also applicable. Therefore, understanding complex-valued functions of a complex argument and their calculus is valuable in quantum mechanics (and will prove to be even more valuable for other physics and representation theory/geometry). Further, Riemann surfaces are a natural sequel. They are a primary basic object of number theory and algebraic geometry too, alongside the Heisenberg representation. Later with Vertex Operator Algebras we will see that they are also a natural setting for quantum field theory ideas.

The final books are special topics within quantum mechanics. Path integrals allow one to formulate the dynamics and observables in terms of classical Lagrangians instead of operator-Hamiltonians, and some problems are more naturally formulated there like the application of classical (e.g. Lorentz) symmetries. Geometric quantization addresses another aspect of the calculus of quantum mechanics left unaddressed, how to consistently form quantum Hamiltonians from classical expressions. The answer turns out to be richer and differential-geometric, the correspondence treats the symplectic structure as a curvature form and constructs the quantum Hilbert space from the underlying classical symplectic manifold. The extra required structures tell us why position and momentum intertwine in the Heisenberg representation, and how quantum mechanics interfaces with geometry. Finally, Connes' book outlines a fundamental generalization of differential geometry where coordinate functions are replaced by operators, leading to a new notion of geometry. Just as in differential geometry, Lie, exterior, and directional derivatives appear in the natural language to recover familiar calculus concepts, in Noncommutative Geometry, the operators detect other geometric information like foliations. He later includes his speculative idea for a Noncommutative Standard model.