# Negative Probabilities

1. Aug 18, 2009

### Dmitry67

I wonder why negative probabilities should be avoided. Except the obvious “it just not possible!”. What happens if we assume that it is possible and just try to go ahead with it?

I suggest looking at it on the MWI platform (even if you don’t like MWI). Why? Because MWI is deterministic, so there are no probabilities at all. We don’t know how probability is defined in MWI. So may be it is bad, but for the discussion of the subject it is good.

So we have a number. We don’t call it a probability. It is >1 or <0? That’s fine! We know that in MWI the feeling or ‘being real’ does not depend on the value of what we call a ‘probability’.

So my question is, except for the fact that it is ‘weird’, are any observable inconsistencies on the microscopic level for the ‘frogs’ in the ‘weird’ branches with ‘probabilities’ outside of 0..1 range?

2. Aug 18, 2009

### Cthugha

In some branches of physics there actually are quasi-probability distributions, which take values smaller than 0.

For example the Wigner quasi-probability distribution in quantum optics behaves in this manner. It is a phase-space distribution and takes negative values for states, which have no classical counterpart. However the negative values usually occur only in small regions and disappear if one tries to take the classical limit. The Glauber-Sudarshan P-representation behaves similarly.

3. Aug 18, 2009

### Dmitry67

Interesting.
So may be Standard Model does not 'fall apart' at High Energies at all?
There are some negative probabilities predicted by SM - but they also dissapear on the classical limt and low energies...

4. Aug 18, 2009

### kote

Dmitry... probability has a specific mathematical definition. Saying probability doesn't have to be bound between 0 and 1 is like saying 1 + 1 doesn't have to equal 2. It's a logical contradiction. Of course, literally any conclusion can be derived from a contradictory premise. This last point is why it can be so appealing (knowingly or unknowingly) to slip an obfuscated contradiction into a theory or explanation. Unfortunately, it just doesn't work.

5. Aug 18, 2009

### gel

I agree completely.

You can have negative numbers, but they aren't probabilities. Even if someone calls them that.

6. Aug 18, 2009

### haushofer

"Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money."

In physics negative probabilities are linked to negative norm states, which means that you need ghostfields to account for that. In a certain way, these negative norm states mean that you have to many degrees of freedom, and adding these ghost fields means basically cutting down the degrees of freedom of your theory.

7. Aug 18, 2009

### Cthugha

This is wrong. You are jumping to conclusions. The mathematical and logical consistency of negative valued probabilities has been shown as early as 1945 by M. S. Bartlett ("Negative Probability". Math Proc Camb Phil Soc 41: 71–73). Feynman and Dirac also proposed the usage of negative probabilities as mathematical tools.

However, the interpretation of these probabilities is of course more complicated. If negative probabilities occur somewhere in QM, this is usually a hint at a state, which does not show up alone, but only as a part of a segregated system or in combination with another state. So in terms of a physical interpretation, you will need to have a look at the total system and combine a negative and a positive probability to some ordinary probability. Negative probabilities can be a sensible mathematical tool and it is just wrong to call them a logical contradiction.

8. Aug 18, 2009

### Dmitry67

Also, it is possible that SOMETHING we got used to call a probability in QM becomes a probability in a low energy limit. If it is true, there will be nothing 'wrong' found in Standard Model at high energies, where negative probabilites start to appear.

9. Aug 18, 2009

### Naty1

You have to qualify what you are trying to accomplish...what's the context. They should clearly be avoided in many contexts, perhaps not in others as above posts suggest.

Kote posted
which is a fine classical answer. But maybe another superior definition will someday be uncovered. In the everyday world we observe, negative probabilities don't exist. You don't see a negative probability in the rolling of dice, for example.

Negative probabilities are an example of many, many mathematical results that do not appear to apply to our everyday world. We have more math than we can use in THIS universe; but if there really are an infinite number of universes, then maybe all our math falls real short....and describes only a small portion of the infinite alternatives.

Galilean transforms don't generally apply because we have relativity and Lorentz Tranforms which match experimental observations; String theory has so far many more particles than we observe; There are many possible formulations of the Einstein Tensor, but the one he picked is supported by experimental observation.

What ARE we to make of all the other math?? I am not sure.

10. Aug 18, 2009

### kote

http://en.wikipedia.org/wiki/Probability_theory
Modern definition: The modern definition starts with a set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by $$\Omega=\left \{ x_1,x_2,\dots\right \}$$. It is then assumed that for each element $$x \in \Omega\,$$, an intrinsic "probability" value $$f(x)\,$$ is attached, which satisfies the following properties:

1. $$f(x)\in[0,1]\mbox{ for all }x\in \Omega\,;$$
2. $$\sum_{x\in \Omega} f(x) = 1\,.$$

That is, the probability function $$f(x)$$ lies between zero and one for every value of x in the sample space Ω, and the sum of $$f(x)$$ over all values x in the sample space Ω is equal to 1. An event is defined as any subset $$E\,$$ of the sample space $$\Omega\,$$. The probability of the event $$E\,$$ is defined as

$$P(E)=\sum_{x\in E} f(x)\,.$$​
If you'd like to redefine probability I suggest you instead just use a different word. You are free, of course, to make up whatever meaning you want for a word. However, using the standard definition, negative probabilities are a logical contradiction.

11. Aug 18, 2009

### kote

There is no such thing as a superior definition of a purely analytic concept. Any difference in meaning necessarily makes a new concept. The fact that we can use the same word to represent varying concepts is an artifact of our language. Are you suggesting that maybe someday someone will come up with a better definition for the word "two"?

Also, as a purely analytic concept, probability is in no way dependent on any conception of universes or any external truths. The concept of probability, as with all mathematical concepts, has been defined a priori. Its entire meaning is contained and represented in its definition. It is not representative of some external object that we can learn more about.

12. Aug 18, 2009

### Dmitry67

Context - Negative probabilities predicted by Standard Model at high energies (LHC energies, not planck/quantum gravity level)

Options:
1. Accept them and try to deal with them. Negative probabilities are an indication that (in a simplest case of the Born rule) square of a probability dencity function is a probability only at the lower energy limit
2. (Widely believed) It means that SM falls apart at high energies and frequire corrections.

13. Aug 18, 2009

### Cthugha

There is no need to redefine it. What you hide in your quote is that you did not quote the definition of probability, but the definition of probability in terms of discrete probability distributions. You will also find another definition of probability in terms of continuous on the very page you quoted. And - most importantly - you will also find the definition of probability in the generalizing case of measure-theoretic probability theory, which also includes strange cases like the Cantor distribution and already differs significantly from the definition in terms of discrete probability distributions. From there, the generalization to negative probabilities is not a huge step. So there is no redefinition, just a generalization. The more general definition includes the one you posted as a limiting case, just as relativity includes classical mechanics as a limiting case.

14. Aug 18, 2009

### kote

I'm pretty sure that what I quoted was very plainly the "modern definition." Anything inconsistent with what I quoted, $$p$$ requires a different definition, $$p'$$. Let $$q$$ be the statement "negative probabilities are impossible."

$$p \rightarrow q$$
$$p' \rightarrow\neg q$$
$$p$$
$$\therefore\neg p'$$

You are arguing that $$p \equiv p'$$, which gives you $$p \land\neg p$$. This is the definition of a contradiction.

Last edited: Aug 18, 2009
15. Aug 18, 2009

### Cthugha

I hate repeating, but the page you quote already has two paragraphs labeled "modern definition". One for discrete probability distributions and one for continuous probability distributions, which are different definitions. Is one more modern and standard than the other or why do you choose one of them? The page also hints at one further definition by telling "The modern approach to probability theory solves these problems using measure theory to define the probability space".

It is clear that a definition of what a probability is can not make any sense without also defining what kind of probability distributions you have in mind. The definition you quoted is fine and the standard definition for probability in terms of discrete distributions, but not for probability in general. This is a huge difference. Discrete distributions are a special case for a special choice of which sample space, algebra, definition of set and measure you use. However, these choices must anyway be adapted to the kind of problem you work on. That many problems lead to discrete probability distributions does not mean that this is the standard definition for all kinds of probability distributions.

Whatever, I think we are already moving far from the original topic.

16. Aug 18, 2009

### kote

You're right. I should have been more specific. You'll notice the continuous definition of probability is also bound by 0 and 1. I chose the discrete definition because it deals with all possible outcomes and events. If we're talking about something being possible or not, we're necessarily not concerned with impossible outcomes. Continuous probability will tell you that there is a pretty good chance you will have between 2.1 and 2.9 children. This is logically impossible, a priori. Discrete probability deals with the complete set of logically consistent outcomes to a situation.

Last edited: Aug 18, 2009
17. Aug 18, 2009

### Cthugha

Right, I agree. In this (and almost all) cases, the given probability distribution deals with all logically consistent outcomes. However there are also situations, where your knowledge of the logically consistent outcomes is necessarily limited and you want to say something about situations, where the logically consistent outcomes are not well defined.

The easiest example is some arbitrary two-dimensional phase space, for example with position and momentum as dimensions. Looking at a classical particles under well defined circumstances, you can easily give the joint probability to find this particle at position x having momentum p. If you now perform a measurement of x alone, you will find, that p(x=x_i) will just be the integral over all possible momentum values p for a certain x_i.

For a quantum particle, the joint probability density does not have a well defined meaning because momentum is not well defined if the position is well known and you are in a position eigenstate. Nevertheless you can find an unambiguous (quasi)probability distribution, which gives you correct results for both marginal distributions in momentum and position (Wigner function). So you get a probability distribution, which gives you correct results for p(x_i) if you integrate over all momentum values at a given x_i and it gives you correct results for p(p_i) if you integrate over all position values at a given p_i. Unfortunately this gives you negative probability values at single points in phase space, but does that matter, if the joint probability density can not be measured (and is not really well defined) anyway?

18. Aug 18, 2009

### kote

I would argue that QM has falsified the idea that position and momentum can both be persistent basic properties of particles. From what I understand this agrees with the Copenhagen and Bohmian interpretations. A particle with simultaneous, classical, position and momentum, is as possible as 2.6 children (assuming QM). The negative probability result is a reductio ad absurdum proof of this. Assume a definite position and momentum (garbage in) and you get negative probabilities (garbage out). It's the same as when I used to screw up on my math homework and end with the equation 1=-1. I didn't disprove logic, I just screwed up in one of my assumptions.

QM absolutely forces us give up a lot about our naive classical views. Sometimes I think the weirdness and obfuscation of the math and theory can cause us to lose sight and go too far. But yeah, as far as I can tell I agree with everything you just posted.

19. Aug 18, 2009

### alexepascual

When two state vectors are orthogonal, we can find the total probability by adding their squared moduli, which in turn represent the sum of the individual probabilities. If they are not othogonal, there will be an interference term which can be negative for destructive interferenct. I guess we could interpret this term as negative probability. But it looks that we can only see negative probability when subtracted from a positive probability and the result of this subtraction can't be less than 0.

20. Aug 18, 2009

### Tac-Tics

I saw this thread hoping there'd be discussion about negative probabilities. Instead, all I found was arguments over nomenclature and the usual QM nonsense.

There is no standard nomenclature for "negative probabilities" because it's a recent idea and because it's not an important area of research (scientists get by with sticking to complex numbers). They aren't the probabilities of classical probability theory, but they are a natural extension of them, just as pseudoriemannian manifolds are a natural extension to riemannian manifolds.

If anyone has any solid information on this study from a mathematical viewpoint (with or without physical applications), I'd be very interested to hear about it. The only reference I've ever seen to this idea comes from Sigfpe's blog:

http://blog.sigfpe.com/2008/04/negative-probabilities.html