# A Amplitudes of probability in Mathematics before QM

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1. Feb 15, 2016

### DaTario

Hi All,

Was there any use of the concept of amplitudes of probability before their use in quantum mechanics?
In connection to this question, who invented or was the first to use this resource?

Best wishes,

DaTario

2. Feb 16, 2016

### mathman

You need to clarify your question. Probability theory goes back several hundred years before quantum mechanics was developed. What do have in mind specifically with "amplitude of probability"? Probability is a defined to be a number.

3. Feb 16, 2016

### DaTario

Ok, l'll try to clarify. A probability amplitude in a discrete sample space context may be a complex number and must be squared to become a well defined probability value. In a continuous sample space context, besides being squared, it must be integrated over some interval in order to typically produce non zero probabilities.

The quantum mechanical method makes use of complex amplitudes, suggesting that, when we are to compute the probability of observing event A or B (disjoint ones) as outcomes, we have to sum their amplitudes and then take the square of this sum to obtain the probability.

This procedure gives birth to what is called interference patterns and that was mainly the cause of its use in QM.

My question is: has anyone (Laplace or some other mathematician) made use of this formalism, i.e., this "way of doing probability calculations", before 1900?

4. Feb 16, 2016

### Hornbein

I don't think there was. Schroedinger was not aware what the amplitudes meant when he came up with his celebrated equation. He was displeased when he found out. I think it was Max Born, grandfather of Olivia Newton-John, who figured it out, but I'm not sure.

5. Feb 16, 2016

### DaTario

Thank you.

I guess the son of Georgine Emilia Brenda knew the answer.

6. Feb 16, 2016

### DaTario

I am afraid that nobody brings these information to students. That quantum mechanics stabilished a new approach in the theory of probabilities. A new way of dealing with the available information when deriving probabilities.
Of course, most teachers explain the method in fair detailed fashion, but it seems to me that we should call attention to these remarkable fact.
Usually, with normal pre QM probability theory, when a new channel is opened for some event occur, the net probability for this event increases.
QM had the urge for a mathematical machinery in which the opening of new channels did not necessarily increase the net probability.

Exemplifying: If I erase the six in a dice and investigate the probability of appearing an even number I will find 2/6 as the probability for there are even numbers, odd number and a blank side as possible outcomes. But if I open a new channel reprinting the six on its due side, the probability will increase to 3/6. It seems that was this aspect that had to be changed to please the QM researchers.

7. Feb 17, 2016

### Number Nine

What does "open a new channel mean"? If six is an option, then the probability is 3/6. If it isn't, then the probability is 2/6. I don't see what QM adds in this case. There's nothing probabilistically groundbreaking about QM -- you square amplitudes and you get probabilities; there's nothing deep or unusual about it (at least, not in a probabilistic sense).

8. Feb 17, 2016

### cosmik debris

Standard probability theory is based on the L1 norm, that is a set of numbers that add to 1. QM is based on the L2 norm which is a set of numbers whose squares add to 1. You could propose systems based on L3 and higher but I believe they don't offer much. Some theorist may have explored these different norms before QM but probably would not have found any use for them in the physical world. So to answer your question; I don't think the idea of probability amplitude existed before QM.

Cheers

9. Feb 17, 2016

### DaTario

Ok, thank you. Let me address a correlated problem. Besides being a theory which is based on either the L1 or L2 norm, the theory of probability has to do with technical procedures for taking profit from available information in order to reduce our uncertanities regarding (typically) the events in nature. So, the theory of probabilities is, in itself, and in some extent, physics. Do you think teachers should stress this fact in introductory QM class, namely, that QM had introduced a new technology and new methods to manipulate the available information in order to reduce our state of uncertainty?

10. Feb 18, 2016

### BvU

Reducing state of uncertainty might not be a justified goal in itself ...

11. Feb 18, 2016

### DaTario

I believe that in some sense it is what really happens. Although the manipulation of information doesn't produce information, in our brains, the change in the linguistic structure may produce a better understanding of what was there in the "message". In this sense, I believe probability theory may be understood as a tool to access coded information in reality (or contexts which resembles reality).

12. Feb 18, 2016

### Stephen Tashi

13. Feb 18, 2016

### Number Nine

I'm not sure what this means. Probability is defined without any reference to a norm; the fact that the numbers "add to 1" just reflects the fact that probability spaces have measure 1. This is still true in QM.

14. Feb 19, 2016

### DaTario

15. Feb 19, 2016

### DaTario

Probability $\rightarrow$ L1
Complex Proability Amplitudes $\rightarrow$ L2

16. Feb 19, 2016

### micromass

Staff Emeritus
What does that even mean? $L^2$-spaces show up plenty enough in classical probability. If you want to talk about variance, then you'll need something in $L^2$.

17. Feb 19, 2016

### DaTario

Ok, That was not my focus. Sorry for not being precise. I was trying to help

I fear you will have to rethink this.
My event was an even number, right?
The throwing of a dice was the general stochastic process in this context.
Without the 6 (let´s change to a five sides dice) there were two channels for my event to appear. Sides 2 and 4. -> probabilty 2/5, right?
By creating a new side, the 6, I have opened a new channel for my event to happen. Probability 3/6, larger.
My point is that, every time in classical probability when there is a change in the structure of the stochastic process, that increases the number of ways (channels) that conduce to the results which corresponds to "my event", its probability suffers an increase as well.
In QM, new channels for a given result may cause destructive interference when you take the square of a sum of complex numbers.

let $z_1$, $z_ 2$ and $z_3$ be three complex numbers such that $0 \le z_1 z_1^* + z_2 z_2^* + z_3 z_3^* < 1$

then, my point in this post is connected to the following mathematical fact:

For certain choices of complex numbers $z_1$, $z_ 2$ and $z_3$:
$(z_1 + z_2) (z_1 + z_2)^* > (z_1 + z_2 + z_3) (z_1 + z_2 + z_3)^*$ .

Best wishes,

DaTario

18. Feb 20, 2016

### Number Nine

Ok, but the word "channel" is not a mathematical or statistical term, so you can't expect anyone to understand what it means unless you define it. Anyway, I'm not sure what relevance this example has to quantum mechanics. Yes, when you increase the number of ways that an event can occur, you increase the probability that it occurs. Does this distinguish probability from QM in some way?

I really don't understand what this has to do with probability theory. In QM, squared amplitudes are interpreted as probabilities. So what? I'm working with reinforcement learning models right now where a program assigns a value $V$ to an action, and $e^V$ gives the probability of selecting that action. Have I invented a new probability theory? One based on exponentials? No, I'm doing perfectly ordinary probability, and the probability of an action just happens to be the exponential of the value assigned to that action. Similarly, QM does perfectly ordinary probability; the probabilities just happen to be given by squared amplitudes. Nothing strange is happening. You keep talking about "channels", but that word doesn't have any mathematical or statistical meaning.

19. Feb 21, 2016

### DaTario

Ok, sorry if you are not familiar with this term, IMO, physicist are somewhat acquainted with that. Mostly because channels are related to modes of propagation of light in some interferometers. I guess you are doing perfectly ordinary probability as long as whenever you increase the number of ways (not channels, right?) for one event to occur your model responds with an increase of the resulting probability. (This may not happen in QM - and if you are not familiar with this fact then I would tell you to try posting questions about this in the appropriate sections of this forum.)
If an event A is well defined and, besides, it corresponds to the occurrence of either A1, or A2 or A3, which are disjoint events, then the "oficial" rule for determining probabilities tells us to sum the probabilities for A1, A2 and A3 in order to obtain the probability for A.
QM orders us to sum not the probabilities, but the complex amplitudes, and only then, to take the square of this sum. I see myself doing something a bit different from what requires perfectly ordinary probability's protocols.
The reason I can understand for one to propose this new way of composition of a probability is the interference which I address in my last calculus (#17).

It is not rare in mathematics to see mathematicians which are familiar with the language of physics.

20. Feb 21, 2016

### Number Nine

Then you should define it. It is not a statistical term, and none of the texts on QM that I've read have used this term in that way. It makes it very difficult to understand what you're asking.

In QM, the squared moduli are interpreted as probabilities, and then ordinary probability theory applies from there. Nothing strange is happening. Nothing.