### A Brief Look at One of John von Neumann's Beliefs about Quantum Mechanics

John von Neumann's argument for quantum logic was based on the Hilbert space formulation of quantum mechanics. In essence a Hilbert space is the quantum analog of phase space used in classical mechanics. In quantum mechanics the basis of interest are not just classical variables like position and momentum but also include spin and super positions of those variables. If there are N possible states a system can be in, then the Hilbert space of that system is the collection of all possible super positions of those N possible states. Where the sub space of a Hilbert space represent the properties of a given quantum system.

Consider the following 2-Dimensional Hilbert space for particle of half integer spin (all given in units of reduced Planck's constant)

Quantum mechanics uses complex, rather than real Hilbert spaces but this will be sufficient to illustrate von Neumann's argument.

In the Hilbert space the component P is a projection operator and operators orthogonal to each other represent "non community operators" which are incompatible properties of a quantum system.

If P is a projector we can represent its negation as

Now simply, the components P refers to a particle with spin in the z-direction.

Whereas its orthogonal component refers to the same particle with a spin in the z-direction of

What's unusual about the Hilbert space that's not corresponded with the phase space in classical mechanics is that it represents other properties too. For example in the diagram above, Q is the particle's spin component in the x-direction

Every ray in the Hilbert space represents some value of the spin of the particle which leads to an obvious problem for quantum mechanics. To understand this lets relabel the positive spin in the z-direction with the label P and the negative spin in the x-direction, Q. How do we represent the conjunction of P and Q? In propositional logic the answer is

In propositional logic the conjunction of two contradictories is false, the problem in quantum mechanics is that it's not even false. The statement is undefined because it cannot be represented on a Hilbert space. Von Neumann and Birkhoff began applying different rules to quantum mechanics than classical physics and instead represented the conjunction of the two bases as the intersection.

What is the intersection? Simple, they only intersect at the zero vector. Now we have meaning to this statement and turns out to agree with propositional logic, the statement is false. Whatever basis you choose it is always orthogonal to the zero vector, so it is always false. The same applies to the conjunction of the negative spin in the z-direction with the positive spin in the x-direction.

In all of these rules of quantum mechanics we encounter a complete contradiction.

We can drop the "x-negative spin or x-positive spin" term given that it's always true. But the left hand side of the equation is always false. Which entails that "z-positive spin" is false. We could easily substitute terms to show that

Is a false statement. Voila! John von Neumann and George Birkhoff showed that quantum mechanics necessitates that orthogonal components are simultaneously true and false. Why would two of the worlds most prominent mathematicians believe in such nonsense? von Neumann and Birkhoff really did believe these calculations were correct but it's probably resolvable, there are a whole variety of different ways of answering their argument.

I'll just consider a couple of them that I find convincing. Previously I talked about the contextuality of quantum measurements, not only is one unable to measure both components of spin simultaneously but the spin of a particle does not exist until measured in some interpretations (Copenhagen, Bohmian mechanics, Many Worlds Interpretation). Further more even if they intersect on the Hilbert space, incompatible properties of a quantum system do not need to be combined if one adopts further rules into quantum theory, such as the single family rule (Consistent Histories). Which I've discussed in previous posts.

Although I reject the case made by von Neumann and Birkhoff I don't want to misrepresent quantum logic. What von Neumann and Birkhoff proposed is one argument for one version of quantum logic. Other versions with far more sophisticated arguments exist.

### William Lane Craig and the Hartle-Hawking No Boundary Proposal

Classical standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984.

These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I&#…

### How Should Thatcherites Remember the '80s?

Every now and again, when I talk to people about the '80s I'm told that it was a time of unhinged selfishness, that somehow or other we learned the price of everything but the value of nothing. I can just remember that infamous line from Billy Elliot; 'Merry Christmas Maggie Thatcher. We all celebrate today because its one day closer to your death'. If it reflected the general mood of the time, one might wonder how it is she won, not one but three elections.

In an era when a woman couldn't be Prime Minister and a working class radical would never lead the Conservative party, Thatcher was both and her launch into power was almost accidental owing in part to Manchester liberals and the Winter of Discontent. Yet I'm convinced her election victory in '79 was the only one that ever truly mattered. Simply consider the calamity of what preceded it, the 1970s was a decade of double-digit inflation, power cuts, mass strikes, price and income controls, and the three…

### Creation Of Universes from Nothing

The above paper "Creation of Universes from Nothing" was published in 1982, which was subsequently followed up in 1984 by a paper titled "Quantum Creation of Universes". I decided it would be a good idea to talk about these proposals, since last time I talked about the Hartle-Hawking model which was, as it turns out, inspired by the above work.
Alexander Vilenkin also explains in a non-technical way the essential idea in his book; Many World's in One – one of the best books I've ever read – it mostly covers cosmic inflationary theory but the 17th chapter covers how inflation may have begun. In fact Vilenkin is one of the main preponderant who helped develop inflation along with Steinhardt, Guth, Hawking, Starobinsky, Linde and others.
Although I won't talk about it here, Vilenkin also discovered a way of doing cosmology by using something called "topological defects" and he has been known for work he's done on cosmic strings, too.
In ex…