Re: Planck Length

[Bruno Marchal]

> On 29 Jan 2019, at 15:03, Philip Thrift  wrote:
> 
> 
> 
> On Tuesday, January 29, 2019 at 5:30:18 AM UTC-6, Bruno Marchal wrote:
> 
>> On 28 Jan 2019, at 15:07, Philip Thrift > 
>> wrote:
>> 
>> 
>> 
>> On Monday, January 28, 2019 at 6:27:37 AM UTC-6, Bruno Marchal wrote:
>> 
>>> On 25 Jan 2019, at 14:53, Philip Thrift > wrote:
>>> 
>>> 
>>> 
>>> On Friday, January 25, 2019 at 6:27:44 AM UTC-6, Bruno Marchal wrote:
>>> 
 On 24 Jan 2019, at 15:19, Philip Thrift > wrote:
 
 
 
 On Thursday, January 24, 2019 at 7:14:15 AM UTC-6, Bruno Marchal wrote:
 
> On 23 Jan 2019, at 19:01, Philip Thrift > wrote:
> 
> 
> 
> On Wednesday, January 23, 2019 at 5:52:01 AM UTC-6, Bruno Marchal wrote:
> 
>> On 22 Jan 2019, at 01:49, Philip Thrift > wrote:
>> 
>> One of the oddest of things is when physicists use the language of 
>> (various) theories of physics to express what can or cannot be the case. 
>> It's just a language, which is probably wrong.
>> 
>> There is a sense in which the Church/Turing thesis is true: All out 
>> languages are Turing in their syntax and grammar. What they refer to is 
>> another matter (pun intended).
> 
> They refer to the set of computable functions, or to the universal 
> machine which understand that language. But not all language are Turing 
> universal. Only the context sensitive automata (in Chomski hierarchy) are 
> Turing universal. Simple languages, like the “regular” one are typically 
> not Turing universal. Bounded loops formalism cannot be either.
> 
> But the notion of language is ambiguous with respect to computability, 
> and that is why I prefer to avoid that expression and always talk about 
> theories (set of beliefs) or machine (recursively enumerable set of 
> beliefs), which avoids ambiguity. 
> For example, is “predicate calculus” Turing universal? We can say yes, 
> given that the programming language PROLOG (obviously Turing universal) 
> is a tiny subset of predicate logic. But we can say know, if we look at 
> predicate logic as a theory. A prolog program is then an extension of 
> that theory, not something proved in predicate calculus.
> Thus, I can make sense of your remark. Even the language with only one 
> symbol {I}, and the rules that “I” is a wff, and if x is wwf, then Ix is 
> too, can be said Turing universal, as each program can be coded by a 
> number, which can be coded by a finite sequence of I. But of course, that 
> makes the notion of “universality” empty, as far as language are 
> concerned. 
> Seen as a theory, predicate calculus is notoriously not universal. Even 
> predicate calculus + the natural numbers, and the law of addition, 
> (Pressburger arithmetic) is not universal. Or take RA with its seven 
> axioms. Taking any axiom out of it, and you get a complete-able theory, 
> and thus it cannot be Turing complete.
> 
> Bruno
> 
> 
> 
> Here's an example of a kind of "non-digital" language:
> 
> More Analog Computing Is on the Way
> https://dzone.com/articles/more-analog-computing-is-on-the-way 
> 
> 
> 
> The door on this new generation of analog computer programming is 
> definitely open. Last month, at the Association for Computing Machinery’s 
> (ACM) conference on Programming Language Design and Implementation, a 
> paper  was 
> presented that described a compiler that uses a text based, high-level, 
> abstraction language to generate the necessary low-level circuit wiring 
> that defines the physical analog computing implementation. This research 
> was done at MIT’s Computer Science and Artificial Intelligence Laboratory 
> (CSAIL) and Dartmouth College. The main focus of their investigation was 
> to improve the simulation of biological systems. 
> 
> 
> Configuration Synthesis for ProgrammableAnalog Devices with Arco
> https://people.csail.mit.edu/sachour/res/pldi16_arco.pdf 
> 
> 
> Programmable analog devices have emerged as a powerful
> computing substrate for performing complex neuromorphic
> and cytomorphic computations. We present Arco, a new
> solver that, given a dynamical system specification in the
> form of a set of differential equations, generates physically
> realizable configurations for programmable analog devices
> that are algebraically equivalent to the specified system.
> On a set of benchmarks from the biological domain, Arco
> generates configurations with 35 to 534 connections and 28
> to 326 components in 1 to 54 minutes.
> 
> 
> - pt
 
 Intersting.
 
 Yet, that d

Re: Histories Of Phenomenally Everything (HOPE)

[Bruno Marchal]

> On 29 Jan 2019, at 21:12, Philip Thrift  wrote:
> 
> 
> 
> On Tuesday, January 29, 2019 at 7:29:43 AM UTC-6, Bruno Marchal wrote:
> 
>> On 29 Jan 2019, at 12:03, Philip Thrift > 
>> wrote:
>> 
>> 
>> This replaces space, time, particles, fields with histories.
>> 
>> I think this is compatible with universal machines.
>> 
>> https://codicalist.wordpress.com/2019/01/28/histories-of-phenomenally-everything-hope/
>>  
>> 
> 
> 
> 
> That space and time, and energy, emerges from histories is compatible with 
> mechanism; even necessary; with mechanism.
> 
> With mechanism you need to convince the universal machine, and the only way 
> too do that, is to let doing the work and discovering this by itself. That 
> has been partially done, and the logic of the observable is a quantum logic, 
> and should be the one that von Neumann and Birkhoff were searching,and which 
> is the one defining all the relative probabilities, imposing a unique 
> measure, like with Gleason theorem.
> 
> A history is not a curve in some space though. A history is defined by a 
> universal number U, its number local data X, and the sequence of steps which 
> follows from U and X (like the phi_U,s(X), s = 0, 1, 2, …).
> 
> Continua and analog situations officers from the first person indeterminacy 
> on all relative computations + the structure imposed by the modal theology of 
> the machine (the intensional variants of auto reference imposed by 
> incompleteness).
> 
> Bruno
> 
> 
> 
> 
> By  "historical paths (curves or walks)", "Histories have a path 
> representation as a sequence" I mean sequences as having a linearly ordered 
> index I  [ https://en.wikipedia.org/wiki/Total_order ], so each element of 
> the history is indexed: 
> 
>  (στ,φ)ᵢ   i ∈ I
> 
> (So can mean sequence as you defined it.)

OK. Just note that with a computation (even with oracle), the sequence is 
determined by the given of a universal machine/number, the input, and then the 
sequence of natural number, playing the role of index steps.

Bruno



> 
> - pt
> 
> 
> 
> -- 
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Re: Coherent states of a superposition

[Bruno Marchal]

> On 30 Jan 2019, at 02:59, agrayson2...@gmail.com wrote:
> 
> 
> 
> On Tuesday, January 29, 2019 at 4:37:34 AM UTC-7, Bruno Marchal wrote:
> 
>> On 28 Jan 2019, at 22:50, agrays...@gmail.com  wrote:
>> 
>> 
>> 
>> On Friday, January 25, 2019 at 7:33:05 AM UTC-7, Bruno Marchal wrote:
>> 
>>> On 24 Jan 2019, at 09:29, agrays...@gmail.com <> wrote:
>>> 
>>> 
>>> 
>>> On Sunday, January 20, 2019 at 11:54:43 AM UTC, agrays...@gmail.com 
>>>  wrote:
>>> 
>>> 
>>> On Sunday, January 20, 2019 at 9:56:17 AM UTC, Bruno Marchal wrote:
>>> 
 On 18 Jan 2019, at 18:50, agrays...@gmail.com <> wrote:
 
 
 
 On Friday, January 18, 2019 at 12:09:58 PM UTC, Bruno Marchal wrote:
 
> On 17 Jan 2019, at 14:48, agrays...@gmail.com <> wrote:
> 
> 
> 
> On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal wrote:
> 
>> On 17 Jan 2019, at 09:33, agrays...@gmail.com <> wrote:
>> 
>> 
>> 
>> On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote:
>> 
>> 
>> On 1/16/2019 7:25 PM, agrays...@gmail.com <> wrote:
>>> 
>>> 
>>> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote:
>>> 
>>> 
>>> On 1/13/2019 9:51 PM, agrays...@gmail.com <> wrote:
 This means, to me, that the arbitrary phase angles have absolutely no 
 effect on the resultant interference pattern which is observed. But 
 isn't this what the phase angles are supposed to effect? AG
>>> 
>>> The screen pattern is determined by relative phase angles for the 
>>> different paths that reach the same point on the screen.  The relative 
>>> angles only depend on different path lengths, so the overall phase 
>>> angle is irrelevant.
>>> 
>>> Brent
>>> 
>>> Sure, except there areTWO forms of phase interference in Wave 
>>> Mechanics; the one you refer to above, and another discussed in the 
>>> Stackexchange links I previously posted. In the latter case, the wf is 
>>> expressed as a superposition, say of two states, where we consider two 
>>> cases; a multiplicative complex phase shift is included prior to the 
>>> sum, and different complex phase shifts multiplying each component, all 
>>> of the form e^i (theta). Easy to show that interference exists in the 
>>> latter case, but not the former. Now suppose we take the inner product 
>>> of the wf with the ith eigenstate of the superposition, in order to 
>>> calculate the probability of measuring the eigenvalue of the ith 
>>> eigenstate, applying one of the postulates of QM, keeping in mind that 
>>> each eigenstate is multiplied by a DIFFERENT complex phase shift.  If 
>>> we further assume the eigenstates are mutually orthogonal, the 
>>> probability of measuring each eigenvalue does NOT depend on the 
>>> different phase shifts. What happened to the interference demonstrated 
>>> by the Stackexchange links? TIA, AG 
>>> 
>> Your measurement projected it out. It's like measuring which slit the 
>> photon goes through...it eliminates the interference.
>> 
>> Brent
>> 
>> That's what I suspected; that going to an orthogonal basis, I departed 
>> from the examples in Stackexchange where an arbitrary superposition is 
>> used in the analysis of interference. Nevertheless, isn't it possible to 
>> transform from an arbitrary superposition to one using an orthogonal 
>> basis? And aren't all bases equivalent from a linear algebra pov? If all 
>> bases are equivalent, why would transforming to an orthogonal basis lose 
>> interference, whereas a general superposition does not? TIA, AG
> 
> I don’t understand this. All the bases we have used all the time are 
> supposed to be orthonormal bases. We suppose that the scalar product (e_i 
> e_j) = delta_i_j, when presenting the Born rule, and the quantum 
> formalism.
> 
> Bruno
> 
> Generally, bases in a vector space are NOT orthonormal. 
 
 Right. But we can always build an orthonormal base with a decent scalar 
 product, like in Hilbert space, 
 
 
 
> For example, in the vector space of vectors in the plane, any pair of 
> non-parallel vectors form a basis. Same for any general superposition of 
> states in QM. HOWEVER, eigenfunctions with distinct eigenvalues ARE 
> orthogonal.
 
 Absolutely. And when choosing a non degenerate 
 observable/measuring-device, we work in the base of its eigenvectors. A 
 superposition is better seen as a sum of some eigenvectors of some 
 observable. That is the crazy thing in QM. The same particle can be 
 superposed in the state of being here and there. Two different positions 
 of one particle can be superposed.
 
 This is a common misinterpretation. Just because a wf can be expressed in 
 different ways (as a vector in the p

Re: Coherent states of a superposition

[agrayson2000]

On Wednesday, January 30, 2019 at 5:16:05 AM UTC-7, Bruno Marchal wrote:
>
>
> On 30 Jan 2019, at 02:59, agrays...@gmail.com  wrote:
>
>
>
> On Tuesday, January 29, 2019 at 4:37:34 AM UTC-7, Bruno Marchal wrote:
>>
>>
>> On 28 Jan 2019, at 22:50, agrays...@gmail.com wrote:
>>
>>
>>
>> On Friday, January 25, 2019 at 7:33:05 AM UTC-7, Bruno Marchal wrote:
>>>
>>>
>>> On 24 Jan 2019, at 09:29, agrays...@gmail.com wrote:
>>>
>>>
>>>
>>> On Sunday, January 20, 2019 at 11:54:43 AM UTC, agrays...@gmail.com 
>>> wrote:



 On Sunday, January 20, 2019 at 9:56:17 AM UTC, Bruno Marchal wrote:
>
>
> On 18 Jan 2019, at 18:50, agrays...@gmail.com wrote:
>
>
>
> On Friday, January 18, 2019 at 12:09:58 PM UTC, Bruno Marchal wrote:
>>
>>
>> On 17 Jan 2019, at 14:48, agrays...@gmail.com wrote:
>>
>>
>>
>> On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal wrote:
>>>
>>>
>>> On 17 Jan 2019, at 09:33, agrays...@gmail.com wrote:
>>>
>>>
>>>
>>> On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote:



 On 1/16/2019 7:25 PM, agrays...@gmail.com wrote:



 On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote:
>
>
>
> On 1/13/2019 9:51 PM, agrays...@gmail.com wrote:
>
> This means, to me, that the arbitrary phase angles have absolutely 
> no effect on the resultant interference pattern which is observed. 
> But 
> isn't this what the phase angles are supposed to effect? AG
>
>
> The screen pattern is determined by *relative phase angles for 
> the different paths that reach the same point on the screen*.  
> The relative angles only depend on different path lengths, so the 
> overall 
> phase angle is irrelevant.
>
> Brent
>


 *Sure, except there areTWO forms of phase interference in Wave 
 Mechanics; the one you refer to above, and another discussed in the 
 Stackexchange links I previously posted. In the latter case, the wf is 
 expressed as a superposition, say of two states, where we consider two 
 cases; a multiplicative complex phase shift is included prior to the 
 sum, 
 and different complex phase shifts multiplying each component, all of 
 the 
 form e^i (theta). Easy to show that interference exists in the latter 
 case, 
 but not the former. Now suppose we take the inner product of the wf 
 with 
 the ith eigenstate of the superposition, in order to calculate the 
 probability of measuring the eigenvalue of the ith eigenstate, 
 applying one 
 of the postulates of QM, keeping in mind that each eigenstate is 
 multiplied 
 by a DIFFERENT complex phase shift.  If we further assume the 
 eigenstates 
 are mutually orthogonal, the probability of measuring each eigenvalue 
 does 
 NOT depend on the different phase shifts. What happened to the 
 interference 
 demonstrated by the Stackexchange links? TIA, AG *

 Your measurement projected it out. It's like measuring which slit 
 the photon goes through...it eliminates the interference.

 Brent

>>>
>>> *That's what I suspected; that going to an orthogonal basis, I 
>>> departed from the examples in Stackexchange where an arbitrary 
>>> superposition is used in the analysis of interference. Nevertheless, 
>>> isn't 
>>> it possible to transform from an arbitrary superposition to one using 
>>> an 
>>> orthogonal basis? And aren't all bases equivalent from a linear algebra 
>>> pov? If all bases are equivalent, why would transforming to an 
>>> orthogonal 
>>> basis lose interference, whereas a general superposition does not? TIA, 
>>> AG*
>>>
>>>
>>> I don’t understand this. All the bases we have used all the time are 
>>> supposed to be orthonormal bases. We suppose that the scalar product 
>>> (e_i 
>>> e_j) = delta_i_j, when presenting the Born rule, and the quantum 
>>> formalism.
>>>
>>> Bruno
>>>
>>
>> *Generally, bases in a vector space are NOT orthonormal. *
>>
>>
>> Right. But we can always build an orthonormal base with a decent 
>> scalar product, like in Hilbert space, 
>>
>>
>>
>> *For example, in the vector space of vectors in the plane, any pair 
>> of non-parallel vectors form a basis. Same for any general superposition 
>> of 
>> states in QM. HOWEVER, eigenfunctions with distinct eigenvalues ARE 
>> orthogonal.*
>>
>>
>> Absolutely. And when choosing a non degenerate 
>> observable/measuring-device, we work in 

Re: Planck Length

[Philip Thrift]

On Wednesday, January 30, 2019 at 5:45:34 AM UTC-6, Bruno Marchal wrote:
>
>
> On 29 Jan 2019, at 15:03, Philip Thrift > 
> wrote:
>
>
>
> On Tuesday, January 29, 2019 at 5:30:18 AM UTC-6, Bruno Marchal wrote:
>>
>>
>> On 28 Jan 2019, at 15:07, Philip Thrift  wrote:
>>
>>
>>
>> On Monday, January 28, 2019 at 6:27:37 AM UTC-6, Bruno Marchal wrote:
>>>
>>>
>>> On 25 Jan 2019, at 14:53, Philip Thrift  wrote:
>>>
>>>
>>>
>>> On Friday, January 25, 2019 at 6:27:44 AM UTC-6, Bruno Marchal wrote:


 On 24 Jan 2019, at 15:19, Philip Thrift  wrote:



 On Thursday, January 24, 2019 at 7:14:15 AM UTC-6, Bruno Marchal wrote:
>
>
> On 23 Jan 2019, at 19:01, Philip Thrift  wrote:
>
>
>
> On Wednesday, January 23, 2019 at 5:52:01 AM UTC-6, Bruno Marchal 
> wrote:
>>
>>
>> On 22 Jan 2019, at 01:49, Philip Thrift  wrote:
>>
>> One of the oddest of things is when physicists use the language of 
>> (various) theories of physics to express what can or cannot be the case. 
>> It's just a language, which is probably wrong.
>>
>> There is a sense in which the Church/Turing thesis is true: All out 
>> languages are Turing in their syntax and grammar. What they refer to is 
>> another matter (pun intended).
>>
>>
>> They refer to the set of computable functions, or to the universal 
>> machine which understand that language. But not all language are Turing 
>> universal. Only the context sensitive automata (in Chomski hierarchy) 
>> are 
>> Turing universal. Simple languages, like the “regular” one are typically 
>> not Turing universal. Bounded loops formalism cannot be either.
>>
>> But the notion of language is ambiguous with respect to 
>> computability, and that is why I prefer to avoid that expression and 
>> always 
>> talk about theories (set of beliefs) or machine (recursively enumerable 
>> set 
>> of beliefs), which avoids ambiguity. 
>> For example, is “predicate calculus” Turing universal? We can say 
>> yes, given that the programming language PROLOG (obviously Turing 
>> universal) is a tiny subset of predicate logic. But we can say know, if 
>> we 
>> look at predicate logic as a theory. A prolog program is then an 
>> extension 
>> of that theory, not something proved in predicate calculus.
>> Thus, I can make sense of your remark. Even the language with only 
>> one symbol {I}, and the rules that “I” is a wff, and if x is wwf, then 
>> Ix 
>> is too, can be said Turing universal, as each program can be coded by a 
>> number, which can be coded by a finite sequence of I. But of course, 
>> that 
>> makes the notion of “universality” empty, as far as language are 
>> concerned. 
>> Seen as a theory, predicate calculus is notoriously not universal. 
>> Even predicate calculus + the natural numbers, and the law of addition, 
>> (Pressburger arithmetic) is not universal. Or take RA with its seven 
>> axioms. Taking any axiom out of it, and you get a complete-able theory, 
>> and 
>> thus it cannot be Turing complete.
>>
>> Bruno
>>
>>
>>
> Here's an example of a kind of "non-digital" language:
>
> *More Analog Computing Is on the Way*
> https://dzone.com/articles/more-analog-computing-is-on-the-way
>
>
>
> *The door on this new generation of analog computer programming is 
> definitely open. Last month, at the Association for Computing Machinery’s 
> (ACM) conference on Programming Language Design and Implementation, 
> a paper  was 
> presented that described a compiler that uses a text based, high-level, 
> abstraction language to generate the necessary low-level circuit wiring 
> that defines the physical analog computing implementation. This research 
> was done at MIT’s Computer Science and Artificial Intelligence Laboratory 
> (CSAIL) and Dartmouth College. The main focus of their investigation was 
> to 
> improve the simulation of biological systems. *
>
>
> *Configuration Synthesis for ProgrammableAnalog Devices with Arco*
> https://people.csail.mit.edu/sachour/res/pldi16_arco.pdf
>
> *Programmable analog devices have emerged as a powerful*
> *computing substrate for performing complex neuromorphic*
> *and cytomorphic computations. We present Arco, a new*
> *solver that, given a dynamical system specification in the*
> *form of a set of differential equations, generates physically*
> *realizable configurations for programmable analog devices*
> *that are algebraically equivalent to the specified system.*
> *On a set of benchmarks from the biological domain, Arco*
> *generates configurations with 35 to 534 connections and 28*
> *to 326 components in 1 to 54 

Re: Coherent states of a superposition

[agrayson2000]

On Wednesday, January 30, 2019 at 2:38:58 PM UTC-7, agrays...@gmail.com 
wrote:
>
>
>
> On Wednesday, January 30, 2019 at 5:16:05 AM UTC-7, Bruno Marchal wrote:
>>
>>
>> On 30 Jan 2019, at 02:59, agrays...@gmail.com wrote:
>>
>>
>>
>> On Tuesday, January 29, 2019 at 4:37:34 AM UTC-7, Bruno Marchal wrote:
>>>
>>>
>>> On 28 Jan 2019, at 22:50, agrays...@gmail.com wrote:
>>>
>>>
>>>
>>> On Friday, January 25, 2019 at 7:33:05 AM UTC-7, Bruno Marchal wrote:


 On 24 Jan 2019, at 09:29, agrays...@gmail.com wrote:



 On Sunday, January 20, 2019 at 11:54:43 AM UTC, agrays...@gmail.com 
 wrote:
>
>
>
> On Sunday, January 20, 2019 at 9:56:17 AM UTC, Bruno Marchal wrote:
>>
>>
>> On 18 Jan 2019, at 18:50, agrays...@gmail.com wrote:
>>
>>
>>
>> On Friday, January 18, 2019 at 12:09:58 PM UTC, Bruno Marchal wrote:
>>>
>>>
>>> On 17 Jan 2019, at 14:48, agrays...@gmail.com wrote:
>>>
>>>
>>>
>>> On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal 
>>> wrote:


 On 17 Jan 2019, at 09:33, agrays...@gmail.com wrote:



 On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote:
>
>
>
> On 1/16/2019 7:25 PM, agrays...@gmail.com wrote:
>
>
>
> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote:
>>
>>
>>
>> On 1/13/2019 9:51 PM, agrays...@gmail.com wrote:
>>
>> This means, to me, that the arbitrary phase angles have 
>> absolutely no effect on the resultant interference pattern which is 
>> observed. But isn't this what the phase angles are supposed to 
>> effect? AG
>>
>>
>> The screen pattern is determined by *relative phase angles for 
>> the different paths that reach the same point on the screen*.  
>> The relative angles only depend on different path lengths, so the 
>> overall 
>> phase angle is irrelevant.
>>
>> Brent
>>
>
>
> *Sure, except there areTWO forms of phase interference in Wave 
> Mechanics; the one you refer to above, and another discussed in the 
> Stackexchange links I previously posted. In the latter case, the wf 
> is 
> expressed as a superposition, say of two states, where we consider 
> two 
> cases; a multiplicative complex phase shift is included prior to the 
> sum, 
> and different complex phase shifts multiplying each component, all of 
> the 
> form e^i (theta). Easy to show that interference exists in the latter 
> case, 
> but not the former. Now suppose we take the inner product of the wf 
> with 
> the ith eigenstate of the superposition, in order to calculate the 
> probability of measuring the eigenvalue of the ith eigenstate, 
> applying one 
> of the postulates of QM, keeping in mind that each eigenstate is 
> multiplied 
> by a DIFFERENT complex phase shift.  If we further assume the 
> eigenstates 
> are mutually orthogonal, the probability of measuring each eigenvalue 
> does 
> NOT depend on the different phase shifts. What happened to the 
> interference 
> demonstrated by the Stackexchange links? TIA, AG *
>
> Your measurement projected it out. It's like measuring which slit 
> the photon goes through...it eliminates the interference.
>
> Brent
>

 *That's what I suspected; that going to an orthogonal basis, I 
 departed from the examples in Stackexchange where an arbitrary 
 superposition is used in the analysis of interference. Nevertheless, 
 isn't 
 it possible to transform from an arbitrary superposition to one using 
 an 
 orthogonal basis? And aren't all bases equivalent from a linear 
 algebra 
 pov? If all bases are equivalent, why would transforming to an 
 orthogonal 
 basis lose interference, whereas a general superposition does not? 
 TIA, AG*


 I don’t understand this. All the bases we have used all the time 
 are supposed to be orthonormal bases. We suppose that the scalar 
 product 
 (e_i e_j) = delta_i_j, when presenting the Born rule, and the quantum 
 formalism.

 Bruno

>>>
>>> *Generally, bases in a vector space are NOT orthonormal. *
>>>
>>>
>>> Right. But we can always build an orthonormal base with a decent 
>>> scalar product, like in Hilbert space, 
>>>
>>>
>>>
>>> *For example, in the vector space of vectors in the plane, any pair 
>>> of non-parallel vectors form a