The lore on complex probabilities

In summary, "The lore on complex probabilities" explores the intricate relationship between probability theory and complex numbers, highlighting how complex probabilities can provide deeper insights into various phenomena in fields such as quantum mechanics and statistical mechanics. It discusses the mathematical foundations of complex probabilities, their applications, and the implications for understanding randomness and uncertainty in complex systems. The text emphasizes the importance of integrating complex analysis with probability theory to enhance predictive models and improve the interpretation of stochastic processes.
  • #1
arivero
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I think that the lore on the need of having probability interference in quantum mechanics and then a complex probability originates in Feynman interpretation of space-time paths, whose probability is weighed with a complex exponential that approaches a dirac delta.

But I can not pinpoint a concrete source of this lore; perhaps it even predated the path integral. Any guesses?
 
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  • #2
AFAIK Dirac mentioned it in 32. The Action Principle, The Principles of Quantum Mechanics.
 
  • #3
Going to check, thanks. It makes sense because Feynman claims to have got is insight from Dirac, I think from some work on contact transformations. But also [I believe to remember that he told...] that he was unable to agree with Dirac on the significance.
 
  • #4
I see, this is the third edition, that already uses Dirac notation:

1707148447124.png


Is it the same in the 1932 edition?
 
  • #5
1930 you mean. It was the 1st edition, second came in 1935, third in 1947, and fourth and final in 1958. I don't have access to any of thr first two editions. Very rare in libraries.
 

FAQ: The lore on complex probabilities

What are complex probabilities?

Complex probabilities extend the concept of probability into the complex plane. Instead of being restricted to real numbers between 0 and 1, probabilities can take on complex values, which have both a real and an imaginary component. This is a theoretical construct used in various advanced fields, including quantum mechanics and certain branches of statistical mechanics.

How are complex probabilities used in quantum mechanics?

In quantum mechanics, complex probabilities arise naturally in the form of probability amplitudes. These amplitudes are complex numbers whose magnitudes squared give the actual probabilities of different outcomes. The complex phase of these amplitudes plays a crucial role in phenomena like interference and entanglement.

Do complex probabilities violate the traditional rules of probability?

While complex probabilities might seem to violate traditional rules, they actually extend them. The square of the magnitude of a complex probability (a real, non-negative number) must still conform to the standard rules of probability, summing to 1 over all possible outcomes. This ensures consistency with classical probability theory when interpreting measurable outcomes.

Can complex probabilities have practical applications outside of theoretical physics?

Yes, complex probabilities can be applied in various fields such as signal processing, control theory, and even finance. In these contexts, they often help model systems with underlying oscillatory or wave-like behaviors, providing a more comprehensive understanding of the system dynamics.

How do you calculate with complex probabilities?

Calculating with complex probabilities involves using the rules of complex arithmetic. This includes addition, multiplication, and taking magnitudes. When dealing with quantum systems, for example, one typically calculates probability amplitudes using the Schrödinger equation and then finds the actual probabilities by taking the magnitudes squared of these amplitudes.

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