Does Quantum Mechanics Need Imaginary Numbers?

Quantum Mechanics
Virtual Numbers in Quantum Mechanics

Concepts put forward in physics have gone through a testing process. Testing these concepts over and over does not change the fact that the concepts are an abstract part. The word concept already contains the abstract. Physicists have pondered this issue for more than a hundred years since quantum truth began to be articulated. In the quantum world, there is inherent uncertainty. A quantum state does not contain enough information to predict the outcome of every possible measurement on the state. Instead, it offers only a probability distribution among the possible outcomes for most measurements. The abstract in the content of our article, "Does Quantum Mechanics Need Virtual Numbers?" We will try to touch on the subject in a simple way.

What is the Uncertainty Principle?

In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of several mathematical inequalities that place a fundamental limit on the accuracy of the values ​​of certain pairs of physical quantities, such as the position of a particle. , x, and momentum, p, can be estimated from the initial conditions.

It is natural to wonder if uncertainty means that quantum mechanics is an incomplete theory. Could it actually be supplemented by a system of local hidden variables that predetermines every measurement result?

Experiments had to be done to work on this idea, thanks to the work of John Bell and others. There was also the question of repeatedly experimenting with the ideas put forward and making it a law.

It can be designed in experiments that distinguish quantum predictions from those of any local hidden variable theory. When experiments are done, quantum theory always comes out victorious. Obviously, the thought formed first, and then the experiment supported it. That's what was interesting.

Another way that quantum theory challenges classical intuition is by involving imaginary numbers.

Observable quantities and probability distributions are necessarily expressed in real numbers.

However, the underlying wave functions and quantum states often contain the non-physical number i, the square root of -1.

But are imaginary numbers a necessary feature of the theory, or are they simply an artifact of its formulation? So is there a completely real formulation that works just as well? It was a concept that was at the forefront of the questions that the physicists of the subject had been thinking about for years.

This is a subtle question.

One can always represent the space of complex numbers as the space of two-dimensional real numbers, one dimension represents the real part of the complex number and the other the imaginary part.

But the quantum states themselves are mathematically represented as multidimensional spaces.

The combination of dimensions over dimensions significantly complicates the mathematics of how quantum systems interact.

Miguel Navascués of the Austrian Academy of Sciences in Vienna and colleagues clarify the question in their new study.

Is it possible to formulate a version of quantum theory that retains some basic mathematical properties of standard quantum theory, but uses only real numbers?

They found that the answer was no. Unfortunately, there was no escape from complex numbers. Actually, it shouldn't be. Now get ready to do some mental gymnastics. Because we have come to the most critical point. Let's expand our somewhat chaotic and paradoxical perspective and focus on the following lines.

Navascués and the researchers devised an experiment that was sketchy in such a way that any real-valued theory would predict different results from standard quantum theory. The experiment is only slightly more extensive than the one used to test Bell's inequality. Now, without going into the details of what has been done, let's briefly explain what the Bell Test is.

What is Bell Inequality Test or Bell Experiment?

A Bell test, also known as the Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Experiments test whether the real world meets local realism, which requires the existence of some additional local variables (called "hidden" because it is not a feature of quantum theory) to explain the behavior of particles such as photons and electrons. All Bell tests to date have found that the local latent variables hypothesis is inconsistent with the way physical systems behave.

Bell Inequality Test
Credit: Adapted from M.-O. Renou et al., Nature 600, 625 (2021)

Now let's consider a line. There are different people on both ends of the line. They transfer information to each other.

Of the two pairs of entangled particles, Bob takes one each, and the other two go to Alice and Charlie.

Bob makes a joint measurement on his particles, and Alice and Charlie each choose from several different measurements to make on their particles.

As with Bell's inequality, theories can be distinguished by their predictions about correlations between measurement results.

But unlike Bell's inequality, it is extremely difficult to calculate expected correlations.

To find an upper bound for real-valued theories, Navascués and his colleagues undertook computation so extensive that it would consume computer memory.

They had to make do with a looser tie than they had hoped.

Still, that's how long it's been since the proposal was made public last January.

Two groups performed the experiment, and both found results in favor of standard complex-valued quantum theory.

It seems that future students of quantum mechanics will have no choice but to grapple with the mathematics of imaginary numbers.

source: physicstoday

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