Complex probability

[m_the0ry]

I have some reading time this summer and, as always, I am thirsting for knowledge. I seek to teach myself quantum physics and math that is 'too theoretical for engineers'. Right now I am reading Feynman as his lectures give the best conceptual understanding of quantum physics despite their age. Still, I can't find a good source for complex probability. If anyone here can personally recommend a complex analysis book that includes probability (or a probability that includes complex analysis, I suppose) I would be thankful.

[Max Raker]

What level have you had complex analysis and probablilty to date? (give the authors names if you can remember) Fenyman is a good place to start for everything though [img]/images/graemlins/smile.gif[/img].

[gumpzilla]

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I have some reading time this summer and, as always, I am thirsting for knowledge. I seek to teach myself quantum physics and math that is 'too theoretical for engineers'. Right now I am reading Feynman as his lectures give the best conceptual understanding of quantum physics despite their age. Still, I can't find a good source for complex probability. If anyone here can personally recommend a complex analysis book that includes probability (or a probability that includes complex analysis, I suppose) I would be thankful.

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I'm not sure what you mean by complex probability. Expand?

Griffith's Introduction to Quantum Mechanics is an excellent book for learning QM.

[PairTheBoard]

[ QUOTE ]
I have some reading time this summer and, as always, I am thirsting for knowledge. I seek to teach myself quantum physics and math that is 'too theoretical for engineers'. Right now I am reading Feynman as his lectures give the best conceptual understanding of quantum physics despite their age. Still, I can't find a good source for complex probability. If anyone here can personally recommend a complex analysis book that includes probability (or a probability that includes complex analysis, I suppose) I would be thankful.

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To be considered "probability" the kind of number that is considered the probability must be a Real Number. You can put Probability measures on a Space of Complex numbers and then compute Real Valued probabilities for Events where the Events are composed of complex numbers. But the probability iteself must be a real number.

In quantum physics they have complex wave functions, which I don't know too much about. But they involve complex numbers. I believe the complex wave function represents our information about the evolving state of some physical thing. It encodes that information in such a way that when we go to observe the physical thing by way of some measurement we can use the complex wave function to produce another kind of Function, which will depend on how we are doing our measurement. The resulting Function is the Probability function which gives Real Valued probabilities for what we are likely to see when we do our measurement.

The Theory works because we can repeat this for many simliar situations and track the statistics for the measurements we get compared to what the Probability Function says we are likely to get.

So calling that "Complex Probability" is a misnomer. You have:

Complex Function --> Probability Function --> checked against statistics for actual measurements.

PairTheBoard

[m_the0ry]

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What level have you had complex analysis and probablilty to date? (give the authors names if you can remember) Fenyman is a good place to start for everything though [img]/images/graemlins/smile.gif[/img].

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He most certainly is, in reading QED I already have a basic understanding of complex probability.


Complex probability is an emergent theory from quantum mechanics. It arises as follows. Quantum theory via Bell's theorem has been proven to be a matter of random variables, thus it is a science of probability. However a problem arises in QM when we try to explain phenomena, most notably interference (which is undeniably part of nature). Probability as a science of its own defines that there is no such thing as a negative probability, and yet in interference, probabilities 'cancel' each other out as if one was negative. The only way this can be explained is in a 2 dimensional plane where each number denotes a positive probability (called a probability amplitude most often) and also phase information. While all probabilities are technically positive and real, phase difference by 180 degrees leads to the 'subtraction' of two probabilities.



My Cpx analysis background is limited and is basically limited to what I need to know to do one directional fourier transforms (time domain to frequency, I can't do vice versa reliably) and elementary complex operations. integrating complex functions and stuff like that are out of my realm.