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RE: [PrimeNumbers] Re: Cracking RSA: Relationship between prime numbers and quantum theory

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  • Paul Leyland
    ... precisely ... You re quibbling. If you prefer it, I m equally happy with occupy exactly . If you really want to get pedantic, I d be even happier with
    Message 1 of 12 , Aug 1, 2001
      > > Not true in general. It is true for fermions (particles with spin
      > > (2i+1)/2) and is known as the Pauli exclusion principle. For bosons
      > > (particles with integral spin) particles can, and do, occupy
      precisely
      > > the same state at the same time. It's why lasers, superfluids and
      > > superconductors have such interesting properties.
      >
      > "occupy precisely" ... can you elaborate? Why don't you use
      > the word "occupy exactly"?

      You're quibbling. If you prefer it, I'm equally happy with "occupy
      exactly". If you really want to get pedantic, I'd be even happier with
      phrasing along the lines of "there is no constraint on the occupancy
      number of an eigenstate of a system of bosons" but that seems unduly
      wordy.

      If we're being pedantic, your original statement is unequivocably false.
      It's not even strictly true for fermions in that the Pauli exclusion
      principle is not really a postulate of quantum mechanics (in the sense
      of a presupposed truth which is not amenable to question) but rather a
      consequence of the anticommutativity of operators acting on fermion
      quantum fields. I was being generous and assumed you meant
      "consequence" or "feature" where you wrote "postulate".

      Even being that generous, your statement "no two particle (sic) can
      occupy the same place at the same time" is false, if by "place" you mean
      spatial location. For a start, two fermions differing only in spin can
      occupy the same energy state. Further, from Heisenberg's uncertainty
      principle, the spatial location of each of two particles can only be
      precisely determined if their momenta are completely undetermined. If
      you know anything about the momenta of the particles, their wave
      functions *will* overlap in space. Trying to nail down "the same time"
      is equally difficult: the particle's energy is then the conjugate
      quantity. But I'll be generous again and assume that by "particle" you
      meant fermion and that by "place" you meant eigenstate.

      If you really want to make progress, I suggest that you consult an
      introductory text or two on quantum field theory. It's 19 years since I
      last studied QFT so the references I can quote from memory are now
      outdated and possibly unavailable, but I'm sure there must be
      contemporary works available.

      (Just checked on Amazon: a search on Quantum Field Theory yields 596
      hits, so you ought to be able to find something. Further, the book I
      own, Elements of Advanced Quantum Theory written by John M Ziman in
      1975, is still in print.)


      Paul
    • Kent Nguyen
      ... First Pauli exclusion principle states: In a closed system, no two electrons can occupy the same state. http://theory.uwinnipeg.ca/mod_tech/node168.html
      Message 2 of 12 , Aug 1, 2001
        On Wednesday 01 August 2001 13:01, Paul Leyland wrote:
        > > > Not true in general. It is true for fermions (particles with spin
        > > > (2i+1)/2) and is known as the Pauli exclusion principle. For bosons
        > > > (particles with integral spin) particles can, and do, occupy
        >
        > precisely
        >
        > > > the same state at the same time. It's why lasers, superfluids and
        > > > superconductors have such interesting properties.
        > >
        > > "occupy precisely" ... can you elaborate? Why don't you use
        > > the word "occupy exactly"?
        >
        > You're quibbling. If you prefer it, I'm equally happy with "occupy
        > exactly". If you really want to get pedantic, I'd be even happier with
        > phrasing along the lines of "there is no constraint on the occupancy
        > number of an eigenstate of a system of bosons" but that seems unduly
        > wordy.

        First Pauli exclusion principle states:
        "In a closed system, no two electrons can occupy the same state."
        http://theory.uwinnipeg.ca/mod_tech/node168.html

        Note Pauli only states for occupance of same state not same time.

        This isn't "exactly" what I'm saying. Go back and read "exactly" what I
        said. Because when you say "occupy exactly" ... I'm very suspicious whether
        you've figured a way to violate Heinsberg uncertainity principle.
        http://www.srikant.org/core/node12.html

        >
        > If we're being pedantic, your original statement is unequivocably false.
        > It's not even strictly true for fermions in that the Pauli exclusion
        > principle is not really a postulate of quantum mechanics (in the sense
        > of a presupposed truth which is not amenable to question) but rather a
        > consequence of the anticommutativity of operators acting on fermion
        > quantum fields. I was being generous and assumed you meant
        > "consequence" or "feature" where you wrote "postulate".

        You don't have to be generous. You need to understand what I wrote. You are
        assuming what I wrote is "Pauli exclusion principle". Having bad assumption
        leads to bad argument.

        >
        > If you really want to make progress, I suggest that you consult an
        > introductory text or two on quantum field theory. It's 19 years since I
        > last studied QFT so the references I can quote from memory are now
        > outdated and possibly unavailable, but I'm sure there must be
        > contemporary works available.

        Working for your employer really makes you *think* you are making progress.
        :)

        >
        > (Just checked on Amazon: a search on Quantum Field Theory yields 596
        > hits, so you ought to be able to find something. Further, the book I
        > own, Elements of Advanced Quantum Theory written by John M Ziman in
        > 1975, is still in print.)

        Thanks for using amazon.com, it's better than bn.com don't you think? :)

        --kent
      • Paul Leyland
        ... No it does not! Just because that web page makes that claim that doesn t mean that the PEP is as stated. The PEP states that no two fermions can occupy
        Message 3 of 12 , Aug 1, 2001
          > First Pauli exclusion principle states:
          > "In a closed system, no two electrons can occupy the same state."
          > http://theory.uwinnipeg.ca/mod_tech/node168.html

          No it does not! Just because that web page makes that claim that
          doesn't mean that the PEP is as stated. The PEP states that no two
          fermions can occupy the same quantum state. Electrons are fermions,
          indeed, but electrons can pair up to form "Cooper pairs" which
          themselves are bosons. These bosons can indeed occupy the same quantum
          state and, when they do, give rise to the phenomenon of
          supercoductivity.

          The web page itself goes on to state "actually, protons and neutrons
          obey the same principle, while photons do not)" something you seem to
          have missed. Lasers function precisely because photons do not obey the
          same principle. Protons and neutrons are spin-half particles and thus
          fermions; photons are spin-zero bosons. Photons, as far as we know,
          have no sub-structure but both protons and neutrons are composite
          particles (as are Cooper pairs and helium nuclei). The helium-4 nucleus
          is a spin-zero boson and so can violate the PEP. When it does, bulk
          helium-4 becomes superfluid. The helium-3 nucleus is a spin-half
          fermion and so liquid helium-3 doesn't become superfluid until the
          temperature is low enough for pairs of nuclei to form spin-zero bosons,
          whereupon it too shows superfluidity.

          > This isn't "exactly" what I'm saying. Go back and read
          > "exactly" what I said.

          Very well, I quote: "One of the postulate in quantum theory states that
          no two particle can occupy the same place at the same time."

          This statement is just plain wrong, for the reasons I went into
          previously.

          > Because when you say "occupy exactly" ... I'm very
          > suspicious whether
          > you've figured a way to violate Heinsberg uncertainity principle.
          > http://www.srikant.org/core/node12.html

          For a start, Heisenberg's uncertainty principle only applies to
          conjugate quantities, such as energy/time and linear momentum/position
          (these two quantities are, of course, special cases of the more general
          4-momentum / spacetime coordinates). It does *not* apply to
          non-conjugate measurements, such as the x-component of momentum and the
          y coordinate, which can be simultaneously measured to arbitrary
          accuracy.

          In general, if the operators corresponding to observables anti-commute,
          HUP applies. If they commute, they do not.

          Please read some real books on quantum theory.


          > assuming what I wrote is "Pauli exclusion principle". Having
          > bad assumption leads to bad argument.

          But that is precisely what you did write!

          > Working for your employer really makes you *think* you are
          > making progress. :)

          I don't think I understand that comment. Don't bother elucidating, as
          the smiley suggests that it's probably not that important.


          I'm becoming ever more convinced that this thread has very little, if
          anything, to do with prime numbers. I've probably already bored the
          majority of readers, so I'll drop out of it here.


          Paul
        • Kent Nguyen
          ... You miss the point of the relation to prime number. Cracking the RSA code is a linear problem, thus a one-dimensional problem. You come and talk about the
          Message 4 of 12 , Aug 1, 2001
            > I'm becoming ever more convinced that this thread has very little, if
            > anything, to do with prime numbers. I've probably already bored the
            > majority of readers, so I'll drop out of it here.

            You miss the point of the relation to prime number.

            Cracking the RSA code is a linear problem, thus a one-dimensional problem.
            You come and talk about the 4th dimension, which to me doesn't seem relevant.
            So you ya, you convince yourself.

            As I've said before, there exist a very close spectra that resemble prime
            number sequence.
            http://www.maths.ex.ac.uk/~mwatkins/zeta/physics1.htm

            My equation with two variables:

            Assume = C1 = P1*P2
            f(x) = x^2 - (P1 + P2)*x + C1 = 0

            I only have one equation with two variables. I need another equation to
            solve for P1 and P2. That's what lead me to quantum mechanic in trying to
            find the wavefunction that describes prime number sequence.

            If P1 = P2, I can use the quadraic formula to solve for x. Resulting in
            sqrt(C1).

            If P1 < P2 or P1 > P2, it's a more difficult situation.

            --kent
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