The Heisenberg Uncertainty Principle states that:
∆px.∆x = ∆E.∆t ≥ ħ/2 .
In other words (as is well known), the uncertainty in a quantum wave-particle’s momentum multiplied by the uncertainty in its position is greater than or equal to half the Dirac constant (Planck’s constant divided by 4π, in effect).
The Uncertainty Principle allows for – strictly temporary – violations of the Law of the Conservation of Energy, whereby ‘virtual particles’ – the mediators or means of exchange of the four forces of nature – are allowed to ‘borrow’ energy from the ‘bank’ of the ‘quantum vacuum’, and ‘pay the loan back’ after a given period of time – the product of the energy and the time being no greater than ħ/2.
Thus ‘virtual gravitons’ are emitted by one massive body and absorbed by another – and you have gravitational attraction between the two bodies; ‘virtual photons’ are emitted by an electron and absorbed by a proton and you have electro-static attraction between them, and so on.
Let me introduce you to my friend, the d’Alembert Operator, named after Jean le Rond d’Alembert (1717-83). It is represented thus:
= (1/c2)(∂2/∂t2) – (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) .
The d’Alembert Operator is the equivalent of the Laplace Operator (the part of the above equation after the minus sign), which is represented by ∇2, and is named after Pierre-Simon Laplace (1749-1827), in Minkowski space-time[1]. The d’Alembert Operator appears in the Klein-Gordon equation, a version of the Schrödinger wave equation that is compatible with the Special Theory of Relativity, unlike Schrödinger’s original, but only applies to spinless particles, unlike the more general Dirac equation.
At its simplest, the Klein-Gordon equation, for a particle with zero charge, but non-zero rest-mass, is:
ψ + m2c2ψ/ħ2 = 0 ,
where ψ is the wave function.
There are a number of things that should be said here. In conventional quantum mechanics, the wave function is not a real function – in fact, literally not, because it is, mathematically speaking, a complex number, taking the form
a ± ib ,
where i = ±√-1, and a and b are real numbers. This complex number, ψ, is said to be a probability amplitude, whose squared modulus is a probability – the probability that a particle will have a location within a given volume of space, or that it will have a given momentum, or a given spin. These probability amplitudes can be expressed as complex vectors in an n-dimensional space (Hilbert space, named after the German mathematician, David Hilbert).
In contrast, in an ordinary wave equation, which is a hyperbolic second-order partial differential equation with no imaginary or complex terms, the wave function f is given by
∇2f = (1/c2)(∂2f/∂t2) ,
which is easily re-arranged as:
f = 0 .
This is, in fact, the Klein-Gordon equation for a spinless particle with zero charge and zero rest-mass. Photons and gravitons have zero charge and zero rest-mass, but photons have spin 1 and gravitons (if they exist – none have been detected so far) have spin 2. There are spinless composite particles, but no elementary ones, as far as we know.
Here, the wave function has the dimension length-squared, or area. How can something that can be as simple as that in one equation be a complicated thing like a ‘probability amplitude’ in another?
I am going to suggest something radical. I am going to suggest we do away with probability amplitudes, and replace them with lengths-squareds. Unfortunately, they won’t be straightforward lengths-squareds, but imaginary ones, and they will involve higher dimensions – not of space, but of time.
We need to go back to Heisenberg, but also to de Broglie. Count Louis de Broglie suggested in 1924 that sub-atomic particles like electrons were associated with waves which would have a wavelength inversely proportional to their momentum, λB = h/mv, where h is Planck’s constant, and m and v the mass and velocity of the particle respectively. It was this idea that set Erwin Schrödinger on the path to his eponymous equation. Let us remind ourselves what that says:
∇2ψ – (2m/iħ)(∂ψ/∂t) – 4πmUψ/ħ2 = 0 ,
where i is ±√-1, again, m and U the mass and potential energy of the particle. The equation is second-order with respect to the spatial co-ordinates, but only first-order with respect to time, so is not, as we have said, compatible with the Special Theory of Relativity – a fault which was rectified by Oscar Klein, Walter Gordon, Paul Dirac, and Ettore Majorana, among others, after January 1926, when Schrödinger published his paper Quantisierung als Eigenwertproblem[2] in the Annalen der Physik, the journal in which Einstein had published his seminal papers on Brownian motion, the photo-electric effect and the Special Theory in 1905.
Schrödinger himself was always convinced that his wave function represented something physically real, and that de Broglie waves themselves were physically real, rather than being ‘waves of probability’ (or rather, of probability amplitude, which makes them even more abstract). However, Marshall Stone and John von Neumann was able to show that wave mechanics – Schrödinger’s system – was equivalent to the matrix mechanics formulated by Heisenberg, Born and
So, how can we improve on this – assuming we can? By introducing a second time dimension – which cannot be a straightforward affair, because if it were, there would be all sorts of untold consequences, but which we unveil here as a means to represent sample space – the space that is normally represented by one axis on a graph of a probability distribution function and indicates all the possible values assumed by a given variable.
This time, however, I am suggesting that ‘sample space’ is something physically real, and not merely a line or space on a piece of graph paper (although that’s real enough, in a way). I am suggesting that it is the space of all possible times that a particle can occupy, so that all of its possible positions, energies, spins and momenta are realised in each one.
I think I can express this idea most simply, thus:
(1/c2)(∂2f/∂t2 + ∂2f/∂σ2) – (∂2f/∂x2 + ∂2f/∂y2 + ∂2f/∂z2) + m2c2f/ħ2 = 0 .
This, of course, simplifies to
(1/c2)(∂2f/∂t2 + ∂2f/∂σ2) –∇2f + m2c2f/ħ2 = 0 ,
which is a version of the Klein-Gordon equation for a spinless, non-zero rest-mass, zero charged particle. In this case, the wave function f is real, and has the dimension length-squared (or area, as above). The dimension σ is given by
σ = ±iλB/c ,
where λB is the particle’s de Broglie wavelength. Consequently, σ2 = –λB2/c2, and
(1/c2)(∂2f/∂t2) – f/λB2 –∇2f + m2c2f/ħ2 = 0 ;
in other words:
f – (f/λB2 + m2c2f/ħ2) = 0 .
Although the equation’s application is very limited, there are – rather more complicated – versions that could be constructed using the same basic principle, namely, that of adding a time dimension.
The idea of travelling sideways in time is one that has fascinated me ever since I saw the Dr Who story, Inferno, with Jon Pertwee as the Doctor, back in the 1970s, and read CS Lewis’s unfinished tale, The Dark Tower, in the 1980s. I also remember reading – I think when I was still in primary school – a book by the American science fiction author, Lester del Rey, called The Infinite Worlds of Maybe, which I re-discovered a few years ago, and is still on my bookshelves.
The Everett de-Witt, or ‘Many Worlds’, Interpretation of quantum mechanics, to my mind, lacks one thing to make it fully plausible, and that is a satisfactory geometry – one that meshes with the metric of space-time of the ΛCDM Model of the Universe. It is not ‘quantum gravity’ that will provide this – this scientific ‘Holy Grail’ is a mirage – but a recognition that a full account of time itself requires that every event not only has spatial co-ordinates, x, y, and z and a linear time co-ordinate, t, but an alternate time, or possibility, co-ordinate, σ.
For an electron in a hydrogen atom, with a de Broglie wavelength of 1.66 × 10-10 m, σ would equal ± i × ~5.545 × 10-19 s. For a macroscopic object, however, say a body weighing 1 kg, travelling at 5 m s-1, its de Broglie wavelength would be 1.325 × 10-34 m, and for it, σ would equal ± i × ~4.42 × 10-43 s! The degree of uncertainty at the macroscopic level is simply too small, when you are dealing with simple, large objects. It’s much different, of course, when you are dealing with complex systems, involving a mutlitude of interacting entities – like the weather, or non-linear dynamical systems. That, however, is a whole different story!
[1] Named, of course, after Hermann Minkowski (1864-1909), Albert Einstein’s Mathematics Professor at the Zurich Polytechnic.
[2] Trans., ‘Quantisation as an Eigenvalue Problem.’ ‘Eigen-’ means ‘quality, property, attribute or adjective.’ ‘Wert’ is ‘worth’, or ‘value’. Clearly, Schrödinger was not talking about property values! For the technical meaning of the term in mathematics, see: http://en.wikipedia.org/wiki/Eigenvalue.
[3] This is because they are vector quantities, with a scalar, or value, part (magnitude) and direction. The value part of these matrices is an eigenvalue; the set of all the eigenvalues is an eigenspace, and the vectors themselves are eigenvectors. Determining a particle’s energy, say, or its position, to any degree of accuracy, entails rendering its momentum indeterminate, and vice versa, precisely because of the non-commutativity of these eigenvectors.
