Quantum Mechanics in the real field

[Telemachus]

Hi there. I had this question going around in my mind for a long time. Basically, I wanted to know if there is a need for the use of the complex field for the wave functions in quantum mechanics, or if quantum mechanics can be built with real wave functions, instead of working in the complex field and the use of the complex field is only driven as a mathematical facility.

As many here know, the physical states in quantum mechanics are described by generalized vectors in Hilbert space, this generalized vectors are complex functions, and the modulus of this complex functions give the probability distribution for the particle position, momentum, etc. The complex wave functions give the facility to work with the 'wave' aspects found in quantum mechanics (they provide the facility to account for the interference terms, destructive and constructive interference, etc). My question is if all of this mathematical facilities provided by the use of the complex field can be recovered by using only real functions instead of complex functions, and if not, which is the root for the need of the complex field in quantum mechanics.

Similarly, one can work in wave classical mechanics with complex functions, by writting the sines and cosines by means of complex exponentials. But then one take the real or imaginary parts to get at the end a real function, which discribes the process in the real world. In quantum mechanics the wave functions are always complex, but the physical predictions are given by the probabilities distributions, related to the modulus of this wave functions, and such modulus are ofcourse always real.

Thanks in advance for your answers,
Regards.

 

[micromass]

Since the complex numbers are built from the real numbers: yes, QM can be made to be entirely real.

 

[DEvens]

To just expand on what micromass said, a tiny bit... You could build the wave function out of two numbers at each location, with special relationships between these two numbers. You would have them define a two component vector with relationships between the magnitude of each component, and transformation relationships between them. And a rule to convert this two component vector into a single real number when you needed an observable value. And rules that meant you could not observe the actual values of the two components, but only the magnitude of the two component vector. And rules such that when the two component vector of this particle and that particle interfere, or the same particle with itself, then the vector addition of the two two-component vectors determined the amplitude for observation.

And you could, steadfastly and even a bit perversely, ignore the fact that these rules were just the rules for complex numbers.

Or you could recognize that ## e^{i\pi}=-1## is a pretty spiffy equation.

 

[ZapperZ]

Similarly, one can work in wave classical mechanics with complex functions, by writting the sines and cosines by means of complex exponentials. But then one take the real or imaginary parts to get at the end a real function, which discribes the process in the real world. In quantum mechanics the wave functions are always complex, but the physical predictions are given by the probabilities distributions, related to the modulus of this wave functions, and such modulus are ofcourse always real.

I don't get this at all!

Since you already know that (i) even in classical mechanics (and, I might add, in electronics as well), we also deal with complex space and (ii) that both classical and QM eventually will deal only with real values, then why are you asking your question as pertained ONLY to QM? Shouldn't you also have the same issues with ALL of physics, electrical engineering, etc... etc? Why pick only on QM when this "problem" is pervasive?

Secondly, I don't quite understand the issues that some people have with complex numbers. What is it about them that make people NOT want to deal with them? It is like there is an inherent aversion to these things, as if they are either bad, evil, or can cause communicable diseases!

Zz.

 

[Telemachus]

I don't get this at all!

Since you already know that (i) even in classical mechanics (and, I might add, in electronics as well), we also deal with complex space and (ii) that both classical and QM eventually will deal only with real values, then why are you asking your question as pertained ONLY to QM? Shouldn't you also have the same issues with ALL of physics, electrical engineering, etc... etc? Why pick only on QM when this "problem" is pervasive?

Secondly, I don't quite understand the issues that some people have with complex numbers. What is it about them that make people NOT want to deal with them? It is like there is an inherent aversion to these things, as if they are either bad, evil, or can cause communicable diseases!

Zz.

Well, in electronics, wave mechanics, etc. it is clear that you always keep the real part, and the mathematics of complex numbers is just a useful artifact to work with. When you define the amplitude of an harmonic oscillator as

##x(t)=A Re [ e^{i\omega t}]##, it is clear that you are working with a real function, using only the complex numbers as an algebraic tool to make things easier.

But in Q.M. the state of the particle is defined in the complex field. Is not that you will keep after all with the real or imaginary parts to get a real function at the end. The state is described by this complex function. Now, the observable is always real, because it is given by the modulus of this complex function. It isn't exactly the same situation at first sight to me. Now that after this discussion it is clear that all this stuff can be worked out as well in the real field, it is clear that complex functions are used only for convenience. But I think that is more subtle stuff in quantum mechanics, than when you work for example with phasors in electric engineering.

I mean, you have lots of relations in the complex field, and theorems, like the Cauchy theorem. All this relations can be taken to the real field? there is actually a necessity for the complex field, starting by the definition of the imaginary unit, isn't it?

 

[Jilang]

Yes, it's just mathematically convenient. Feynman considered real rotating phases in his little arrows.

 

[bhobba]

Now, the observable is always real, because it is given by the modulus of this complex function. It isn't exactly the same situation at first sight to me.

Complex numbers are absolutely necessary to QM. If you don't have complex numbers you do not get path cancellation in Feynman's sum over histories approach and you do not get the principle of least action.

See the following:
http://arxiv.org/pdf/1204.0653.pdf

Thanks
Bill