Axioms of Quantum Semiotic

Peder Voetmann Christiansen
IMFUFA, RUC

The sign relation

  1. The quantum mechanical state vector is a sign.

  2. A sign or representamen (R), according to Peirce, is a first standing in such a genuine triadic relation to a second, called its object (O), as to be capable of determining a third, called its interpretant (I), to assume the same triadic relation to its object in which it stands itself to the same object.(note 1)

  3. The representamen R in a quantum semiotic sign relation mediates between the quantum mechanical object O and the interpretant I:
    I-R-O

  4. The interpretant I is a potential, actual, or general purely physical result of measurement.

  5. The sign links ( - ), in the dyadic parts R-O and I-R of the sign relation are interaction bonds corresponding to the physical processes of preparation (the R-O link) and registration (the I-R link).

  6. Each sign link is characterized by the Peircean categories as either 1: potential, 2: actual, or 3: general.

  7. The category numbers, f and g, of the R-O link and the I-R link are restricted by the selection rule: g f.

  8. The qualisign 11 (g = f = 1) is the continuum of the Hilbert space H. The symbol (g = f = 3) is synthesized from the lower signs by successive actualizations of potential links (1 2) and generalizations of actual links (2 3).

  9. The six classes of signs (gf) are connected with Peirce's semiotic definitions and Dirac's bra-ket notation in the following way
    (33)
    symbol
    qp
    (13)
    iconic legisign
    p
    (23)
    indexical legisign
    p
    (11)
    qualisign
    H
    (12)
    iconic sign
    (22)
    indexical sinsign

The measurement process

  1. A measurement is a permanent registration. The physical setting of an interpretant (the I-R link) preceding the registration is an irreversible process.

  2. Registration is a dissipative and noisy process.

  3. For a dissipative admittance the quantum noise on the current, whose spectrum is given by the fluctuation- dissipation (FD) theorem (note 2) corresponds to a time-series of discrete events

  4. For a dissipative device with mobility and relaxation time the average number of events up to time t following an event at time 0 at zero temperature according to the FD theorem is given by

  5. The collapse or reduction of the state vector requires the setting of a dissipative sign link corresponding to the appropriate ray of H before the measurement. The projection on the ray is the first of the quantum events predicted by the FD theorem. (note 3)

  6. The collapse of a state vector for more than one particle requires prospective coincidence counting.

  7. The violation of Bell's inequalities and other superclassical correlations is due to a common context of detection of several particles represented by preset coincidence counters.

  8. Quantum Mechanics is strictly local and all the so called "non-local" effects can be simulated in a purely classical and local scenario provided there is a common context for the registration of individuals. (note 4)

Notes:

  1. Collected Papers, ed. Hartshorne & Weiss, CP 2.274.
  2. H. B. Callen and T. A. Welton, Phys. Rev., 83, 34 (1951).
  3. P. V. Christiansen, The Semiotics of Quantum-Non- Locality, IMFUFA text no. 93 (1985).
  4. See my paper "Peircean local realism does not imply Bell's inequalities", Joensuu 1990.
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