Is it possible to do quantum mechanics without complex functions

[cooev769]

Given that the wave equation can be represented in complex form for simplicity sake, for normalising by multiplying by complex conjugate and adding amplitudes and so forth.

Would it be possible to do everything with only real wave functions instead of complex wave functions. Given we rederived the Schrödinger equation and so forth using the real wave function instead of the complex one?

And if not why not? I thought the complex wave function was only for use of convenience and nothing more.

 

[Vanadium 50]

In general, you have the choice of using one complex function or two coupled real ones. For simplicity, we normally choose the former.

 

[cooev769]

Oh so it can be done? Awesome.

 

[Fredrik]

What V50 is saying is that you can use two real-valued functions instead of one complex-valued function. I wouldn't interpret that as "it can be done".

"The logic of quantum mechanics" by Beltrametti and Cassinelli spends a few pages on a discussion of the possibility of using a vector space over $\mathbb R$ instead of a vector space over $\mathbb C$. If I understand page page 246 correctly (I'm not sure that I do), there's an issue with including rotations in this theory that will force you to use a pair of functions instead of just one function. You can then use those two functions to define a complex-valued function, which lives in a complex vector space. But what we get this way isn't the usual quantum theory of a single spin-0 particle, because in this theory, the class of operators that represent measuring devices in the real world is larger than it should be. In addition to the self-adjoint linear operators, this class also contains the self-adjoint $\mathbb R$-linear operators, i.e. all self-adjoint T such that T(af+bg)=aTf+bTg for all a and b such that I am a=Im b=0.

 

[Vanadium 50]

It's been known since the 1st half of the XIXth Century that you can replace a differential equation in a complex variable with four differential equations in two complex variables.

Most people do not regard this as an improvement and therefore stick with the complex representation.

 

[akhmeteli]

Given that the wave equation can be represented in complex form for simplicity sake, for normalising by multiplying by complex conjugate and adding amplitudes and so forth.

Would it be possible to do everything with only real wave functions instead of complex wave functions. Given we rederived the Schrödinger equation and so forth using the real wave function instead of the complex one?

And if not why not? I thought the complex wave function was only for use of convenience and nothing more.

Please see https://www.physicsforums.com/showpost.php?p=4337679&postcount=4 and other posts in that thread. Let me emphasize that what I discuss there is not the trivial replacement of complex numbers by pairs of real numbers.

 

[micromass]

From a mathematical point of view, the complex numbers are just defined as pairs or reals. So everything that can be done with complex numbers can be done without them. The theory will just be less elegant.

In the same way, everything that can be done with rational numbers can in principle be done with real numbers. So you don't even need real numbers. Then again, would you really want a physical or mathematical theory that refuses to work with $\sqrt{2}$, $\pi$ or $e$?? It is possible, but it is surely not going to be insightful, easy or elegant.

 

[Matterwave]

In the same way, everything that can be done with rational numbers can in principle be done with real numbers. So you don't even need real numbers. Then again, would you really want a physical or mathematical theory that refuses to work with $\sqrt{2}$, $\pi$ or $e$?? It is possible, but it is surely not going to be insightful, easy or elegant.

Do you mean the converse? Everything with real numbers can be done with rational numbers? Since rational numbers is a subset of the reals, it's trivial that everything that can be done with rational numbers can be done with real numbers...

Also, if you meant the former, could you explain a little of how you would get irrational numbers of out rational numbers? I'm guessing you would use some infinite sequence of rational numbers or some such? I have not encountered this thought before, and it seems interesting.

 

[micromass]

Do you mean the converse? Everything with real numbers can be done with rational numbers? Since rational numbers is a subset of the reals, it's trivial that everything that can be done with rational numbers can be done with real numbers...

Yes, sorry.

Also, if you meant the former, could you explain a little of how you would get irrational numbers of out rational numbers? I'm guessing you would use some infinite sequence of rational numbers or some such? I have not encountered this thought before, and it seems interesting.

Well, I haven't really thought this out very well, but since the reals can be defined from the rationals, it is clear that we can just rephrase everything that needs real numbers into statements only using rationals. So I would not use irrational numbers at all, but just use sequences of rational numbers to simulate them.

All measurements that we can possibly do only involve rational numbers. For example, if I draw a straight line and try to measure it with a ruler, I can never measure the length of the line perfectly accurately. The only thing I can say is that the length is between $15$ mm and $16$ mm. So everything we really need are rational numbers, since measurements only make rational numbers. It's not that we can actually accurately measure $pi$, for example. Furthermore, things like circles are also only an abstraction that doesn't exist in reality.

So what we represent by irrational numbers now can be represented by (Cauchy) sequences of real numbers. The interpretation of this sequence would then be a improving sequence of observations that get better and better.

This won't yield any different physics. The current physics can all be restated in this context, although it would look very ugly and unmotivated.

But I don't want to derail this thread. If you're interested in a further discussion, then PM me and I'll make a new thread about this.

 

[UltrafastPED]

Everything with real numbers can be done with rational numbers?

Every real number is defined in terms of a set of rational numbers; the process is called "making a Dedekind cut". This is not the only way to construct the real numbers; the goal is to make a 1-1 correspondence with the concept of a geometric line ... hence the real number line.

See http://en.wikipedia.org/wiki/Dedekind_cut

and for more methods see:
See http://en.wikipedia.org/wiki/Construction_of_the_real_numbers

Or you can work backwards from the hyperreals:
http://en.wikipedia.org/wiki/Hyperreal_number

I like the hyperreals ... makes infinities and infinitesimals feel warm and fuzzy. :-)

 

[anorlunda]

There is more to it than simplicity in mathematics. The results are very different when the quantities combined are complex.

I recommend Richard Feynman's famous book QED. Using clever little graphical illustrations, Fenyman explains the role of the complex phase without any mathematics at all. Your library might have the book.

 

[micromass]

There is more to it than simplicity in mathematics. The results are very different when the quantities combined are complex.

There shouldn't be any difference. Whether you use complex numbers, or pairs of real numbers or rotation matrices, the difference is just aesthetic. The math results won't change one bit what kind of representation you use. Hence the physics won't change either. You won't get different results.

 

[bhobba]

There shouldn't be any difference

That is correct - but it's much more elegant with complex numbers - you would have to have rocks in your head to do it as coupled reals.

Why complex numbers? Its got to do with having continuous transformations between pure states.

The argument goes something like this. Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Standard probability theory is basically the theory of mixed states where the pure states describe the elements of some event space. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense out of these weird complex pure states.

Thanks
Bill