# Tag Archives: reality

# Math, physics and reality

This blog has been nice. It doesn’t get an awful lot of traffic (about a thousand visitors a week) but, from time to time, I do get a response or a question that fires me up, if only because it tells me *someone* is actually *reading *what I write.

Looking at the site now, I feel like I need to reorganize it completely. It’s just *chaos*, right? But then that’s what gets me the positive feedback: my readers are in the same boat. We’re trying to make sense of what physicists tell us is reality. The *interference model *I presented in my previous post is really nice. It has all the ingredients of quantum mechanics, which I would group under two broad categories: uncertainty and duality. Both are related, obviously. I will not talk about the *reality *of the wavefunction here, because I am biased: *I* firmly believe the wavefunction represents something real. Why? Because Einstein’s E = *m*·*c*^{2} formula tells us so: energy is a two-dimensional oscillation of mass. Two-dimensional, because it’s got *twice *the energy of the classroom oscillator (think of a mass on a spring). More importantly, the real and imaginary dimension of the oscillation are both real: they’re perpendicular to the direction of motion of the wave-particle. Photon or electron. It doesn’t matter. Of course, we have all of the transformation formulas, but… Well… These are *not *real: they are only there to accommodate *our *perspective: the state of the observer.

The distinction between the *group *and *phase *velocity of a wave packet is probably the best example of the failure of ordinary words to describe reality: particles are not waves, and waves are not particles. They are both… Well… Both at the same time. To calculate the *action *along some *path*, we assume there is some path, and we assume there is some particle following some path. The path and the particle are just figments of our mind. Useful figments of the mind, but… Well… There is no such thing as an infinitesimally small particle, and the concept of some one-dimensional line in spacetime does not make sense either. Or… Well… They do. Because they help *us *to make sense of the world. Of what *is*, whatever it is. 🙂

The mainstream views on the physical significance of the wavefunction are probably best summed up in the Encyclopædia Britannica, which says the wavefunction has no physical significance. Let me quote the relevant extract here:

“The **wave function****, **in quantum mechanics, is a variable quantity that mathematically describes the wave characteristics of a particle. The value of the wave function of a particle at a given point of space and time is related to the likelihood of the particle’s being there at the time. By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave (or de Broglie wave), although for such waves amplitude has no physical significance. The square of the wave function, Ψ^{2}, however, does have physical significance: the probability of finding the particle described by a specific wave function Ψ at a given point and time is proportional to the value of Ψ^{2}.”

Really? First, this is *factually *wrong: the probability is given by the square of the *absolute* value of the wave function. These are two *very *different things:

- The square of a complex number is just another complex number: (a
*+ i*b)^{2 }= a^{2 }+ (*i*b)^{2 }+ 2*i*ab = a^{2 }+*i*^{2}b^{2 }+ 2*i*ab = a^{2 }– b^{2 }+ 2*i*ab. - In contrast, the square of the absolute value always gives us a
*real*number, to which we assign the mentioned physical interpretation:|a*+ i*b|^{2 }= [√(a^{2 }+ b^{2})]^{2}= a^{2 }+ b^{2}.

But it’s not only position: using the right *operators*, we can also get probabilities on momentum, energy and other physical variables. Hence, the wavefunction is so much more than what the Encyclopædia Britannica suggests.

More fundamentally, what is written there is *philosophically *inconsistent. *Squaring *something – the number itself or its norm – is a mathematical operation. How can a mathematical operation suddenly yield something that has *physical* significance, if none of the elements it operates on, has any. One cannot just go from the mathematical to the physical space. The mathematical space *describes *the physical space. Always. In physics, at least. 🙂

So… Well… There is too much nonsense around. Disgusting. And the Encyclopædia Britannica should not just present the mainstream view. The truth is: the jury is still out, and there are many guys like me. We think the majority view is plain wrong. In this case, at least. 🙂