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|“I want to know how God created this world. I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts; the rest are details.”
– Albert Einstein
Back in Chapter 18, we introduced a special type of Vector Space pertinent to Particle Physics, the Lie Algebra. In this Chapter, we will cover another special Vector Space, one of these that is at the very heart of Quantum Dynamics, the Hilbert Space.
Hilbert Spaces are named after the German Mathematician David Hilbert, whose image appears above and who, incidentally, was the first person to use the word eigen in connection with -vector and -values. However Hilbert’s contributions extended beyond both of these achievements; he was unarguably one of the greatest figures in Mathematics during his lifetime and his influence continues to this day .
As with many topics I touch on in this book, Hilbert Spaces would require a volume of their own to cover them in any great detail. Here I will skim the surface, establishing enough of a feel for us to later consider their role in Particle Physics.
At least on a cursory examination, a Hilbert Space is somewhat familiar. It is a Vector Space occupying the less recherché quarters of this extensive conurbation; Vector Spaces resembling Euclidean Space. The reader will recall that, when journeying to a general (and more abstract) definition of a Vector Space, we started from Euclidean Space. Unlike some Vector Spaces that can be constructed, in Euclidean Space it means something to ask both “how long is a vector?” and “what is the angle between two vectors”. Hilbert Spaces are more general versions of Euclidean Space in which these questions can still be asked and answered.
Back in Chapter 15, we introduced the dot product of two vectors in Euclidean Space, a binary operator which includes information about both the size of vectors and the angles between them. This concept can be extended to non-Euclidean Spaces by the concept of an Inner Product, we will deal with what this means as part of the overall definition of a Hilbert Space below.
Hilbert Spaces can have many more dimensions than 3D Euclidean Space and indeed one of the motivations for them is to formalise infinite-dimensional analogues of Euclidean Space, in particular allowing for the paraphernalia of the Calculus to be meaningfully applied to them. More formally:
A Hilbert Space is a Vector Space, V, together with an inner product, a related norm, which supports a distance function (making it a Metric Space), that is also complete.
We will first spend a bit of time on both the inner product and an associated norm. Both concepts relate to everyday experience. Let’s start by considering the inner product, which as mentioned above is related to the dot product in Euclidean Space.
For a Vector Space, V, over a Field, F (which in this case can be either ℝ or ℂ but not some other general Field), the Inner Product of two vectors, v and u is written 〈v, u〉 and is a binary operator:
〈., .〉 : V × V ↦ F
with the following properties:
- For all v, u ∈ V:
〈v, u〉 = 〈u, v〉
Where – as normal – the bar indicates complex conjugation.
This is known as Conjugate Symmetry
- For all v, u, w ∈ V and a ∈ F:
a〈v, u〉 = 〈av, u〉
〈v + u, w〉 = 〈v + w〉 + 〈u + w〉
This property applies only to the first argument of the inner product and is thus called Linearity in the first argument 
- 〈0, 0〉 = 0
and for all non-zero v ∈ V:
〈v, v〉 > 0
This is described as the inner product being Positive Definite.
Both the norm of a Hilbert Space and the distance function it defines can be described in terms of the inner product:
For a Vector Space, V, over a Field, F (which in this case can be either ℝ or ℂ but not a general Field), the Norm of a vector v is written ||v|| and is a mapping from V to F such that
||.|| : V ↦ F : v ↦ √〈v, v〉
The norm enables a distance function, d(v, u), for any two vectors which is defined as the norm of their difference, so:
d(v, u) = ||v – u||
This must have the following properties:
- For all v, u ∈ V:
d(v, u) = d(u, v)
I.e. the distance between v and u is the same as the distance between u and v.
- For all v ∈ V:
d(v, v) = 0
and for all all v, u ∈ V, such that v ≠ u:
d(v, u) > 0
I.e. the distance between a vector and itself is 0 and that between two distinct vectors is non-zero. This is another example of something being Positive Definite.
- For all v, u, w ∈ V:
d(v, u) ≤ d(v, w) + d(u, w)
Which is known – by obvious analogy with Euclidean Space – as the Triangle Inequality and states that it takes longer to go from A to B via a third point C than directly.
The inner product, the norm, the distance function it defines and the hand-wavy concept of completeness together yield a Hilbert Space.
At this point, I would normally provide a example to solidify this concept and / or provide some perspective as to the meaning of Hilbert Spaces. Both of these things will appear later in the book, but the examples tend to require some concepts we are yet to cover and the meaning that we are interested in, the one pertaining to Quantum Mechanics, will have to wait for the closing part of the book.
[The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy.
In the mathematically rigorous formulation of quantum mechanics, developed by John von Neumann, the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex separable Hilbert space, known as the state space, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each observable is represented by a self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.
The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.
For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.]
[To be completed]
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Chapter 22 – Notes
Not least via his famous list of 23 Hilbert Problems, articulated in a speech to the 1900 International Congress of Mathematicians, and which have formed something of a manifesto for many areas of Mathematical research ever since. Solving a Hilbert Problem is akin to receiving a Fields Medal (the “Nobel Prize” for Mathematicians), though the two are not mutually exclusive (the late Paul Cohen is a member of both sets for example). At the time of publication, 3 of the 23 Problems remain unsolved, 8 have been partially addressed and a couple are not really problems, but instead calls for further work. Problem 8 is one of the outstanding 3, something I touched on myself recently.
All I will offer is an analogy based on work we did back in Chapter 4. We know that ℚ ⊂ ℝ. It can also be shown that – in a loose sense of the word – the Real Numbers are more dense that the Rational Numbers. The Reals fill in gaps in the Rationals, which contain none of √2, e or π, for example; whereas no numbers intersperse the Reals. A complete space is like the Reals in that it has no gaps to be filled.
Completeness is more formally defined in terms of the convergence of limits of certain series. As we explored in Chapter 20, convergence of limits is a cornerstone of the Calculus.
This may be contrasted with linearity in both arguments, or bilinearity, something we met in Chapter 18.
Text: © Peter James Thomas 2016-17.
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