Amplitudes of probability in Mathematics before QM

[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

 

[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.

 

[DaTario]

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.

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?

 

[Hornbein]

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

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.

 

[DaTario]

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.

Thank you.

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

 

[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.

 

[Number Nine]

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.

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).

 

[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

 

[DaTario]

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

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?

 

[BvU]

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

 

[DaTario]

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

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).

 

[Stephen Tashi]

This abstract doesn't pre-date QM, but it suggests that there can be purely mathematical connections between probabilities and complex numbers: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2046444

 

[Number Nine]

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

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.

 

[DaTario]

This abstract doesn't pre-date QM, but it suggests that there can be purely mathematical connections between probabilities and complex numbers: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2046444

It seems that the author is not working with the complex probability amplitudes (those which, squared, give rise to probabilities).

 

[DaTario]

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.

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

 

[micromass]

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

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 $L^2$.

 

[DaTario]

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 $L^2$.

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

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).

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. -> probability 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

 

[Number Nine]

Ok, That was not my focus. Sorry for not being precise. I was trying to helpI 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. -> probability 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.

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?

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

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.

 

[DaTario]

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?

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).

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.

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

 

[Number Nine]

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.

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.

QM orders us to sum not the probabilities, but the complex amplitudes, and only then, to take the square of this sum.

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

 

[chiro]

I think the OP is trying to understand why the square of these quantities yields the probability as opposed to its consistency.

In other words - the OP wants to understand why you can find the amplitude of a complex number and it represents the actual probability itself.

Most people are saying that you can define probabilities in any way that meet the Kolmogorov axioms (basically) but I think the OP wants to know the reason for the definition as opposed to the definition itself being mathematically consistent (with said Kolmogorov axioms).

 

[DaTario]

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

Does Classical probability presents desctructive interference?

Besides, I have mentioned QM explicitly in the OP. If it is not a familiar theme for you, you shouldn't have tried to enter the discussion.

Best wishes

DaTario

 

[fresh_42]

If it is not a familiar theme for you, you shouldn't have tried to enter the discussion.

Instead of being rude couldn't you as well have defined what 'channel' to you mean? I only know it in the context of information theory as well.
And sorry for entering the discussion. Some of us are here to learn something.

 

[DaTario]

I think the OP is trying to understand why the square of these quantities yields the probability as opposed to its consistency.

In other words - the OP wants to understand why you can find the amplitude of a complex number and it represents the actual probability itself.

Most people are saying that you can define probabilities in any way that meet the Kolmogorov axioms (basically) but I think the OP wants to know the reason for the definition as opposed to the definition itself being mathematically consistent (with said Kolmogorov axioms).

Fine, Ok that QM's way of computing probabilities is consistent with Kolmogorov Axioms. The most important point here is the specific property relating the probability of A OR B to occur (assuming A and B are disjoint events). Classicall probability tells us to sum the probabilities of A and B, while quantum mechanics tells us to sum the complex amplitudes os A and B and then to take square of this sum. Mathematically the second way brings a new phenomena to the probability theory, called destructive intereference. OP puts the following question: have this method of complext amplitudes been used before QM or in contexts not directly related to QM?

 

[DaTario]

Instead of being rude couldn't you as well have defined what 'channel' to you mean? I only know it in the context of information theory as well.
And sorry for entering the discussion. Some of us are here to learn something.

Dear fresh 42, I was not being rude. I was trying to put some limit in the rude pressure that participant was doing on me.

And, yes, the word channel in quantum optics has been captured from the context of information theory.

 

[DaTario]

After some reflection about the comments I received, I decided that this post is done for me and I would like to appologize to you all for not succeed in being clear.

Thank you

Best Regards,

DaTario

 

[Demystifier]

Does Classical probability presents desctructive interference?

It does not. See
http://lanl.arxiv.org/abs/quant-ph/0505143

 

[Demystifier]

Was there any use of the concept of amplitudes of probability before their use in quantum mechanics?

No there wasn't.

In connection to this question, who invented or was the first to use this resource?

The physicist Max Born.

 

[DaTario]

Thank you,

Although I couldn´t grasp all the paper's content, it is there pretty clear that such interference effects are not present is his classical approach.

Best wishes,

DaTario