Bimodal Quantum Theory ∗ In process Saurav Dwivedi† Incepted. 25.APR.2011 Last Processed December 14, 2011 m w w w . g e o c i t i e s . w s / d w i v e d i / d a t a / b q t . p d f Work in Progress Abstract Some variants of quantum theory theorize dogmatic "unimodal" states-ofbeing, and are based on hodge-podge classical-quantum language. They are based on ontic syntax, but pragmatic semantics. This error was termed semantic inconsistency [1]. Measurement seems to be central problem of these theories, and widely discussed in their interpretation. Copenhagen theory deviates from this prescription, which is modeled on experience. A complete quantum experiment is "bimodal". An experimenter creates the system-under-study in initial mode of experiment, and annihilates it in the final. The experimental intervention lies beyond the theory. I theorize most rudimentary bimodal quantum experiments studied by Finkelstein [2], and deduce "bimodal probability density" π = |ψin〉 ⊗ 〈φfin| to represent complete quantum experiments. It resembles core insights of the Copenhagen theory. keywords. Pragmatism; Bimodal Logic; Probability. PACS. 03.65.Ud; 05.30.Ch; 02.50.Cw Contents 1 Introduction ii 1.1 The System Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 2 Quantum Disconnections and Acausality iii 2.1 Bimodal Prescription and Vertex Model . . . . . . . . . . . . . . . . iv 2.1.1 Quantum Topology . . . . . . . . . . . . . . . . . . . . . . . . iv 2.2 Quantum Jumps and Acausality . . . . . . . . . . . . . . . . . . . . . v ∗A rudimentary form of this work was presented at a "Talk" given at Max-Planck-Institut für Mathematik in Die Wissenschaften, Leipzig, Germany 16.AUG.2011 †k Saurav.Dwivedi@gmail.com m www.geocities.ws/dwivedi i 1 Introduction S. Dwivedi 3 Quantum Probability v 4 Probability Density Operator vii 4.1 Unimodal Mathematical Expectation . . . . . . . . . . . . . . . . . . x 5 Complementarity x 6 Experimental Quantification xi 6.1 Experimental Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . xii 6.2 Bimodal Mathematical Expectation . . . . . . . . . . . . . . . . . . . xii 7 Experimental Phase xiii 8 Wave Mechanics and Objectivity xiii 8.1 Objects and Frame Relativity . . . . . . . . . . . . . . . . . . . . . . xiii 8.2 States of Being . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 9 Correspondence xv 10 Summary xvi 1 Introduction The name "Quantum Mechanics" carries the germ of mechanistic philosophy. Many people endure to fit quantum theory into this name. They seem to work on causal or ontic interpretations of quantum theory. Practical quantum theory is not mechanistic. It resembles Bohm's anti -Bohmian proposals [3]. We can not reduce an individual quantum system-under-study to finite and coherent set of quantitative entities, that define its qualitative infinity [3]. A mechanistic theory suffices this requisite. Classical mechanics is a mechanistic theory, where coherent quantitative set (q, p) represents qualitative diversity of the system-under-study. Quantum theory has no such representatives. The doctrine with name "Quantum Mechanics" tacitly entails mechanistic prescription, that deceptively raises paradoxes from experiments that are incompatible with mechanistic prescription. We change the prescription to avoid paradoxes, which seems neater way to reduce ambiguity. Our prescription is pragmatic, bimodal and non-mechanistic, which obviates paradoxes. I call a practical quantum theory "Pragmatic Relativity", in that its syntax is pragmatic, and it relativizes absolute "states-of-being" [4]. Pragmatic relativity is not a variant of Copenhagen theory. Mechanistic theories are modeled on states-of-being. The "state-of-being" is unimodal dogmatic entity. An experimenter measures state-of-being of the system-understudy solely in initial mode of the experiment. A final mode seems redundant, in that it measures the same state. Classical experiments are unimodal idealistic measurements, that envisage states-of-being of the system-under-study. States-of-being objectivize the system-under-study, and entail its ontology. Copenhagen theory deviates from this prescription. It has its roots in the philosophy of Niels Bohr. It appeared to him when he thought "What does it mean to know ( ii - ) Bimodal Quantum Theory 1.1 The System Interface the atom, while we see it only when it changes". He brought Heraclitean tendency into physics. I recapitulate the Copenhagen theory, theorizing most rudimentary bimodal quantum experiments, and deduce "bimodal probability density" π = |ψin〉 ⊗ 〈φfin| to represent complete quantum experiments. 1.1 The System Interface Quantum theories model experiments on a miniscule part of Cosmos - a systemunder-study S - a microcosm. An isolated system-under-study is oxymora; we see a system while we interact with it. This prescription respects temporal locality; a quantum system is "local" immediate connection with the experimenter. There is no "global" isolated system. Quantum experimenter E (who lives in exosystem, E ⊂ XS) divides Cosmos into the dichotomy; endosystem S and exosystem XS, creating the system interface or experimental channel S |XS. Experiment destroys this interface1, being sole process that connects S with XS. Experiment entangles system S with exosystem XS . Quantum experiment is system-episystem entanglement S –XS. An experimenter creates the system-under-study in initial mode of experiment, and annihilates it in the final. We do not theorize the system-under-study apart from these two pragmatic events. Quantum systems jump from initial mode to final mode of experiment, without carrying a dogmatic entity that defines its reality after the experiment. The initial system does not evolve into the final in intervention of the experiment. In classical continuous experiments, an auxiliary "system interface" separates the system from being effected by the experimenter; the system-under-study carries dogmatic state-ofbeing that we seem to predict causally after the experiment. Classical experimental channel is unidirectional interface; system-under-study acts on the experimenter, but not conversely. These experiments are often termed "ideal experiments", and deceptively preempted by ontologists in the discussion of quantum theories. We see classical systems as they are. There is no interface in quantum experiments, and actions respect reciprocity. Classical systems are dogmatic unimodal objects. Quantum systems are bimodal pragmatic events. In classical experiments, the system-understudy is an absolute object. In quantum experiments, the experimenter is an absolute subject. A theory based on relative experimenter lies beyond Copenhagen theory, and being developed elsewhere [4]. 2 Quantum Disconnections and Acausality Causality is succession of events: how an event descends from the preceding one. An event is an act or happening, like initial and final modes of an experiment. Einstein 1This interface was arbitrary in Heisenberg's version of Copenhagen theory, but Bohr's insistence eliminates it [1]. Heisenberg refrained from eliminating this split between observer and the systemunder-study, in order to preclude the dilemma of considering whole universe as system-under-study [5, Ch. VII.1]. Heisenberg retained interface in order to save the observer. Rendering a theory of whole cosmos non practical, I retain Bohr's proposals. ( iii - ) 2.1 Bimodal Prescription and Vertex Model S. Dwivedi called collision of two bodies an "event". He ascribed "space-time" address to his event; a dogmatic event. In our prescription, an event is pragmatic; the happening itself (verb), than its address (noun). Von Neumann's theory [6], though rendered a variant of Copenhagen theory [2], is unimodal. It theorizes initial mode vector ψ (or ket |ψ〉 in Dirac's terminology). Inferences from past experiments endow the system-under-study with states-of-being, while predictions for future experiments are statistical. This theory seems to have temporal asymmetry; past experience represents the system-under-study causally, while future predictions do not. Dirac's theory [7] is also unimodal, that ascribes the system-under-study an state-of-being |ψ〉. The error lies in their modal structure that we cure next. 2.1 Bimodal Prescription and Vertex Model Quantum theory2 advances modal logic. Classical experiments were unimodal; a classical experimenter prepared and registered the system-under-study simultaneously. This simultaneity ascribes to the system its unimodal "states-of-being". Classical experimenter measures state-of-being of the system-under-study. Quantum theory renounces this simultaneity and unimodality. Quantum experiments are bimodal; a quantum experimenter injects, prepares or creates the systemunder-study in its initial mode, and extracts, registers or annihilates it in the final mode of experiment. Quantum experimenter specifies bimodal external acts, not unimodal internal states. Quantum experiment is succession of external acts on the system-under-study. We can only define it by its initial and final modes. Initial mode creates the system-understudy by sending a probe, and interacting with it. The system-under-study undergoes a drastic irreversible change; we call it "intervention" of the experiment. The experimenter is not supposed to know of the intervention; an attempt to know it ends the experiment, that we call its final mode. Final mode annihilates system-under-study. We know (and can know) the system-under-study only while interacting with it, or acting upon it; in initial and final modes the experiment. Quantum knowledge is both bimodal and pragmatic. A quantum experiment can be described by at least two modes; initial and final, with no causal connection. 2.1.1 Quantum Topology Quantum topology is theory of quantum connections; how quantum events do connect. Processes such as creation and annihilation are quantum events. Definition 2.1. A quantum system Q is represented by two bimodal pragmatic vertices Qin and Qfin , connecting system-under-study with the experimenter E. The initial vertex Qin represents initial mode of experiment, and final vertex Qfin represents the final mode. Qin represents creation of the system-under-study and Qfin its annihilation. The archetypal action diagram of a "complete" quantum experiment is give by 2By quantum theory, I tacitly mean Copenhagen theory, unless otherwise explicated. ( iv - ) Bimodal Quantum Theory 2.2 Quantum Jumps and Acausality Arrows EQin and EQfin represent initial and final actions |ψin〉 and 〈φfin| of episystem E on the system Q . The intervention I is a disconnected event, that lies outside the theory. The system-under-study jumps from Qin to Qfin , at least for the experimenter E. Actions connecting pragmatic vertices of Q with E constitute experimenter's frame FE{ψi, φf ; τ} ; where ψ and φ represent initial and final external actions of his choices, i and f represent their indices, and τ represents "proper time" of experimenter's clock. ψ or ket | 〉 represents creator, and φ or bra 〈 | represents annihilator. Experimental frame FE{ψi, φf ; τ} belongs to episystemic composite frame space In⊗ Fin⊗Time ; |ψin〉 ∈ In , 〈φfin| ∈ Fin , τ ∈ Time . 2.2 Quantum Jumps and Acausality Quantum systems "jump" from Qin to Qfin with no causal connection. A complete bimodal quantum experiment connects Qin and Qfin to experimenter E . The experimenter does not know how Qin evolved into Qfin . His knowledge is restricted merely to the mode vectors of his choices randomly drawn from his frame FE{ψi, φf ; τ} . Quantum events Qin and Qfin are disconnected, and we can not entail causality to assimilate their evolution. This discontinuity was often called quantum jump and endured by ontologists to refute. It appears that quantum jumps are indispensable, and can not be renounced while defining reality. We are eventually led to probabilistic prescription; talk about what is possible, than real. Many people blame quantum theory for not being causal. Some disqualify it, and work on its causal or ontic variants, that lack modal semantics; this compendium is abundant. Heuristically, I entail acausality a symptom of more elaborate prescription for theorization. Unimodal dogmatic theories, having mechanistic syntax, are causal ; bimodal pragmatic theories are not. 3 Quantum Probability Quantum probability was erroneously called charge density, when probability was confused with (electron) density. Born later resolved this problem in his probabilistic interpretation of quantum theory [8], and generalized charge density e|ψ|2 of electrons in atoms to more general quantum systems. Contemporary quantum theories are modeled on density operator ρ = |ψ〉 ⊗ 〈ψ| (a variant of Pauli's "probability amplitude"). However, density operator ρ = |ψ〉 ⊗ 〈ψ| (often called density matrix) is not Schrödinger probability operator; it does not fulfill criterion of Schrödinger operators, and does not correspond to his eigenvalue equation. It was erroneously ( v - ) 3 Quantum Probability S. Dwivedi called a quantum operator because it was represented by matrices. It seems a pseudo Schrödinger operator, in that P = Tr (ρ) . Nevertheless, I retain terms density operator or statistical operator for it. I theorize a probability ω to account for system's ontology, and call it mode probability. It represents modes-of-being of the system-under-study. When we set out observation on the system-under-study, we may or may not observe all its characteristic (or systemic) variables or coordinates proper to the experimenter E or his frame FE{ψi, φf ; τ} . The system-under-study might fall in one of these modes-of-being: • Apparent mode The system-under-study may be rendered in apparent mode of being when all its variables are specifiable at one instance (in experimenter's frame), and system-under-study could be specified (or objectivized) uniquely by means of these systemic variables. [Theories modeled on apparent mode perception are relic of ontology.] • Partial mode The system-under-study may be rendered in partial mode of being when some of its variables are specifiable at one instance (in experimenter's frame), and system-under-study could not be specified with ultimate precision by means of these variables. Systems in partial mode of being are maximalinformative-systems (in language of Von Neumann) or simply quantum systems. [Practical quantum theories are usually modeled on partial mode perception, which are pragmatic in usage; deceptively ontic in form.] • Hidden mode The system-under-study may be rendered in hidden mode of being when none of its variables are specifiable at one instance (in experimenter's frame), and system-under-study could not be specified by any means. These modes can alternatively be called Full, Possible and Null modes of being. Quantum mode is possible mode, not full (or apparent) being. Quantum mode is maximal informative mode. Systemic variables are often complementary. Two complementary variables, say (q, p) entail phase or state space of classical systems. In quantum usage, knowledge of one precludes that of another. Classical systems or objects are systems in apparentmode-of-being; with specification to all their systemic variables. Quantum systems are maximal-informative-systems, in partial-mode-of-being; with specification to few of their systemic (complementary) variables, or all within minimal uncertainty set out by Heisenberg [9]. Some endure to supplement quantum systems with some "hidden" noncomplementary variables, that render it in apparent mode-of-being, and anticipate that we would eventually recede to classical epistemology. These hidden-variables could not be subjected to perception, as their name implies, and these endeavors lead nowhere. We (can) specify the system-under-study (only) by means of its mode-of-being. I entail some systems in practice: • Objects If the system-under-study is in apparent-mode-of-being, and an experimenter E learns (or measures) all its systemic variables, we call the systemunder-study an object. [It is not in that all experimenters agree upon its appearance. They change it when then attempt to perceive it. They independently, ( vi - ) Bimodal Quantum Theory 4 Probability Density Operator and rather separately render it an object. Our object differs from ontological "absolute" objects (like Moon). See Section 8.1] • Maximal-informative-systems If the system-under-study is in partialmode-of-being, and an experimenter E learns (or measures) only few of its systemic variables, or all within minimal uncertainty implied by uncertainty relations [9], we call the system-under-study a maximal-informative-system or quantum system Q. • Forbidden-systems If the system-under-study is in forbidden-mode-of-being, and an experimenter E does not (or can not) learn (or measure) any of its systemic variables, we call the system-under-study a forbidden-system. [Vacuum is a forbidden system.] In quantum usage, we confront with maximal-informative-systems. Experiments perceiving maximal-informative-systems many be called maximal informative experiments or simply quantum experiments. 4 Probability Density Operator Despite its rudimentary form S = K lnψ [10], Schrödinger's wavefunction is often written nowdays in polar form [11] ψ(R, S) := R exp ( i } S ) . (4.1) Its partial differentiation w.r.t action S yields ψ(R, S) + i} ∂ψ(R, S) ∂S = 0 . (4.2) Definition 4.1. A hypothetical "toy" identity operator I (without physical meaning), in analogy to identity matrix I, corresponds to Schrödinger eigenvalue equation (SE) Î|ψ〉 = I|ψ〉 = |ψ〉 with I ≡ I ≡ 1 , SE : I −→ Î . (4.3) In view to Def. 4.1, (4.2) yields Schrödinger identity operator Î = −i} ∂ ∂S . (4.4) I theorize, phenomenologically, most rudimentary bimodal quantum experiments studied by Finkelstein (1996) [2]; Malus-Born experiment. Our system-under-study is a beam of "photons" traversing optical arrangement of Malus experiment. The archetype of the experiment is absorb←− analyze←− polarize←− emit . The experimenter has a frame of experiment FE{ψi, φf ; τ}, that constitutes of his choices for initial and final mode vectors (or polarization vectors) ψin and φfin ; ( vii - ) 4 Probability Density Operator S. Dwivedi where i and f represent indices of initial and final mode vectors. [Caution! φfin is distinguished from ψin , and both represent physically distinguished actions (i.e., ψin 6= φfin). ψin and φfin belong to different mathematical spaces called initial space and final space. Finkelstein studies their operational symmetry [2]. Here, ψ (and ket | 〉) is preempted for initial and φ (and bra 〈 |) for final mode vectors. bra 〈 | represents modal dual (MD) of ket | 〉 for arbitrary mode vectors, MD : | 〉 −→ 〈 | , and φfin may not be confused as modal dual of ψin , except for ψ ≡ φ .] Extreme classical cases are with φfin ‖ ψin and φfin ⊥ ψin ; former passes the photon, while later blocks it. Quantum phenomena occurs with oblique polarizers φfin ∠ ψin . For oblique polarizers, Malus calculated the fraction of photons transmitted, that he found to be P = cos2 ∆θ , (4.5) where ∆θ is angle difference between polarizer |ψin〉 and analyzer 〈φfin| . His fraction P was rediscovered by Born, that he called transition probability [2]. The generalized form of Malus-Born transition probability is P = |〈φfin|ψin〉|2 . (4.6) For Malus experiment, ψin and φfin are both normalized: 〈ψfin|ψin〉 = 〈φfin|φin〉 = 1 , where ψfin is modal dual of ψin and φfin is modal dual of φin . Modal dual of polarizer is analyzer with same polarization angle θ . The analyzer 〈φfin| with same polarization angle θ (i.e., ψ ‖ φ) as polarizer |ψin〉 transmits the photon. ψin and φfin are not orthogonal necessarily, except for φ ⊥ ψ, 〈φfin|ψin〉 = 0 ; orthogonal polarizer and analyzer "block" the photon. In either case, ψin and φfin are not normalized or orthogonal simultaneously; they do not form orthonormal bases for Malus experiment. Trace (Tr) of Schrödinger "toy" identity operator (4.4), Tr (Î) = 〈φfin|Î|ψin〉 = 〈φfin|ψin〉 , (4.7) returns amplitude of Malus-Born experiment. Definition 4.2. Bimodal probability density π = |ψin〉 ⊗ 〈φfin| represents a binary composite action, creation⊕ annihilation, of experimenter E on the system-understudy. It is mere notational, and order of ψin and φfin is irrelevant. It represents a complete quantum experiment proper to experimenter's frame FE{ψi, φf ; τ} . Caution! It may not be confused with unimodal density matrix ρ, which is ρin = |ψin〉〈ψin| for initial mode, and ρfin = |φfin〉〈φfin| for final mode of experiment. Unimodal quantum theories are modeled on "initial density matrix" or "initial statistical operator" ρin , not final ρfin . Trace (Tr) of bimodal probability density π , Tr (π) = 〈φfin|π|ψin〉 = |〈φfin|ψin〉|2 , (4.8) returns transition probability (4.6) of Malus-Born experiment. [Caution! For unimodal density operators, Tr (ρin;φ) = 〈φfin|ρin|φin〉 = |〈φfin|ψin〉|2 and Tr (ρfin;ψ) = 〈ψfin|ρfin|ψin〉 = |〈φfin|ψin〉|2 . It owes semantic error, and makes no experimental sense; we preempt bimodal density operator π = |ψin〉〈φfin| and bimodal trace Tr (π;ψin, φfin) = 〈φfin|π|ψin〉 , instead.] ( viii - ) Bimodal Quantum Theory 4 Probability Density Operator Definition 4.3. ω is probability of mode-of-being of the system-under-study, or simply mode probability. For a system-under-study in apparent-mode-of-being, ω = 1. For a system-under-study in hidden-mode-of-being, ω = 0. For quantum systems in partial-mode-of-being, ω ∈ (0, 1) . For ∆θ = π/4 , Malus calculated P = 1/2. This seems paradoxical for beam consisting of one photon. It makes an absurd assertion "half of the photon passes through analyzer". For one photon, ω ∈ (0, 1) ; an individual quantum system is maximal-informative-system with mode probability ω ∈ (0, 1) . For extreme classical cases, ∆θ = 0 or π/2, we know whether a photon passes or not e.g., ω = 1 or 0 . For ∆θ = 0 and ω = 1, a photon is apparent (or objective); experimenter knows that it passes the analyzer. For ∆θ = π/2 and ω = 0, it is forbidden; experimenter knows that it does not pass the analyzer. For ∆θ ∈ (0, π/2) and ω ∈ (0, 1) , experimenter does not know the system-under-study at individual level. ω represents possibility (or potential) for a quantum system, despite extreme classical cases (ω = 1 or 0). We should better call ω quantum probability henceforth. (4.5) makes sense for many photons. Yet we do not know which photon passes, and which does not. We can not know quantum systems at individual level. Quantum theory is many system theory. For three different cases: φfin ‖ ψin , φfin ∠ ψin and φfin ⊥ ψin , (4.8) yields Tr (π) = 1 for parallel polarizers φfin ‖ ψin ; Tr (π) ∈ (0, 1) for oblique polarizers φfin ∠ ψin ; Tr (π) = 0 for perpendicular polarizers φfin ⊥ ψin . (4.9) In view to Def. 4.3, (4.8) and (4.9), Tr (π) corresponds to quantum probability ω , Tr : π −→ ω , ω = Tr (π) = 〈φfin|π|ψin〉 = |〈φfin|ψin〉|2 . (4.10) Bimodal probability density π represents "potential" for mode probability ω . Transition probability P is tacitly equivalent to mode probability ω . Recall (4.7) and Def. 4.2 for unimodal quantum theories, where ψ and φ form orthonormal bases, and ME {Â} = Tr (Â) for an arbitrary Schrödinger operator Â . A trivial case in Malus experiment with polarizer |ψin〉 and analyzer 〈ψfin| with same polarization angle θ (∆θ = 0) resembles unimodal quantum theories. Here, Tr (Î) = ME (Î) = 〈ψin|Î|ψin〉 = ρin . We preempt ρ ≡ Î , and dispense with "toy" identity operator. We endow ρ with physical semantics that Î lacked. We have Schrödinger probability density eigenvalue equation ρ|ψ〉 = ρ|ψ〉 , (4.11) with Schrödinger "unimodal probability density operator" ρ = −i} ∂ ∂S . (4.12) Its unimodal trace (Tr) returns "unimodal probability density" ρ = Tr {ρ} , ρ = |ψ〉 ⊗ 〈ψ| . (4.13) ρ = −i} ∂/∂S and ρ = |ψ〉〈ψ| have little utility in bimodal quantum theories, than π = |ψin〉〈φfin| . ( ix - ) 4.1 Unimodal Mathematical Expectation S. Dwivedi 4.1 Unimodal Mathematical Expectation For a predicate P of the system-under-study, "unimodal mathematical expectation" (ME) is given by P def= ME {P} = Tr (Pρ) Tr (ρ) = Tr (P ρ) Tr (ρ) = 〈ψ|P|ψ〉 〈ψ|ψ〉 , (4.14) where P is Schrödinger operator of predicate P ; ME : P −→ P . Here ρ = −i} ∂/∂S and ρ = |ψ〉⊗〈ψ| ; ψ (4.1) is initial mode vector, often deceptively envisaged unimodal dogmatic statevector of the system-under-study. For pure mode vectors ρ2 = ρ ; for mixture of mode vectors ρ2 ≤ ρ ; and Tr (ρ) = 1 for both. 5 Complementarity between Quantum Probability and Ψ-Wave For an arbitrary initial modevector |ψ〉, such as (4.1), unimodal mathematical expectation (ME) of commutator [Ŝ, ρ]− yields ME {[Ŝ, ρ]−} = 〈[Ŝ, ρ]−〉 = 〈ψ|[Ŝ, ρ]−|ψ〉 = i}〈ψ|ψ〉 , (5.1) or ME : [Ŝ, ρ]− −→ i} , 〈[Ŝ, ρ]−〉 = i} . (5.2) [Caution! Our deduction follows from unimodal quantum theories, where ket |ψ〉 and bra 〈ψ| are both initial mode vectors, and orthonormal. We follow mathematical expectation criterion of Von Neumann's [6] or Dirac's [7] unimodal quantum theories, not from Section 6.2] In view to generalized uncertainty relation σ2Sσ 2 ρ ≥ ( 〈[Ŝ, ρ]−〉 2i )2 , we obtain σSσρ ≥ } 2 or ∆S∆ρ ≥ } 2 , (5.3) where ∆ measures spread in S and ρ . Quantum unimodal probability density (ρ) and Action (S) are complementary; knowledge of one precludes that of another. For an apparent system-under-study or object (with ρ = 1)3, action S is ambiguous, and does not represent property of the system. For a partial or quantum system-under-study (with ρ ∈ (0, 1)), S is known with definite precision (5.3). In either case, S is an episystemic variable, and represents experimenter's action on the system-under-study. In classical theory, S was functional of the path, not function of the system. S represents action of episystem on the system, not systemic variables. 3In unimodal quantum theories, quantum probability ω = Tr (ρ) . Roughly ρ = √ ω ; ρ = 1, 0 for ω = 1, 0 and ρ ∈ (0, 1) for ω ∈ (0, 1) . ( x - ) Bimodal Quantum Theory 6 Experimental Quantification We adduce, in view to (4.1), (4.10) and (5.3), Quantum probability ω and ψ function are complementary. Knowledge of one precludes that of another. ψ , being a function of S , represents action of episystem on the system, and ω , trace of ρ , represents quantum probability. ψ constitutes experimenter's frame FE{ψi, φf ; τ} , and represents no property of the system-under-study at all. Complementary entities are not apparent simultaneously. For an apparent systemunder-study (one with ω = 1), ψ is partial. 'Apparent', 'proper' and 'characteristic' are synonymous, at least in our context. Apparent systems do not have characteristic or eigen ψ. Apparent systems (or absolute objects) have states of being, but not the ψ. Statements like "eigenvector |ψ〉 of the system-under-study" or "state function ψ of the system-under-study" are oxymoronic. A statement like "ψ is maximal informative function for the system-under-study" is more relevant for what quantum experiments ascertain. Nevertheless, we could know ψ (4.1) during initial and final modes of quantum experiments (with ω ∈ (0, 1)), with definite precision controlled by (5.3). It may not be assimilated in that ψ represents the system-under-study partially, incompletely or with finite precision; instead, we see the system-under-study as a counterpart of the experimenter, not as an object in its own essence. For isolated systems or objects, ψ (4.1) is ambiguous; uncertainty diverges ∆S,∆ψ →∞ . Knowing and doing are complementary. Schrödinger conceived ψ (4.1) as "particle like wavefunction" to reconcile waveparticle duality and path discreteness of particles in Wilson cloud chamber [10]. Bohr called particle and wave complementary modes of the system-under-study. ψ (4.1) can not be known from particulate systemic variables alone in the sense of complementarity. Von Neumann (1932) adduced that we can not describe quantum systems causally even though we know their wavefunction. For him, state variables (q, p) and wavefunction ψ are essentially different structures [6, Ch. III]. To describe system's statesof-being, one needs to supplement ψ with additional parameters that were called hidden. Hidden parameters were preempted by classical mechanicians to reduce statistical relations to causal ones, for example, in kinetic theory of gases. It became philosopher's stone of physics, and many endured in vain to search it, including Enstein [12], Bohm [13, 14] and many others. Von Neumann adduces that to reduce quantum theory from statistical to causal interpretation, by means of supplementing hidden parameters, is impossible [6, Ch. IV.2]. He renders statistical interpretation of Born as only consistent interpretation of quantum theory. 6 Experimental Quantification Quantification transcends one system theories to many. Here, I quantify experiment, not the system-under-study. Unimodal initial density operator ρin = |ψin〉〈ψin| represents "partial" or incomplete experiments. It also lacks modal structure, and owes ( xi - ) 6.1 Experimental Assembly S. Dwivedi semantic error [2]. So does the final density operator ρfin = |φfin〉〈φfin| [15]. A complete quantum experiment has (at least) two modes; initial mode injects, prepares or creates the system-under-study, and final mode extracts, registers or annihilates it. The intervention lies beyond the theory. Bimodal density operator π = |ψin〉〈φfin| represents an individual "complete" experiment on the system-under-study. Several quantum experiments are represented by an assembly of bimodal density operators πe , e representing index of the experiment. The experimenter E creates and annihilates the system-under-study in several trials proper to his frame FE{ψi, φf ; τ} ; i and f index initial and final mode vectors of his choices. Each experimental "trial" entails a unique bimodal density operator πe = |ψine 〉 ⊗ 〈φfine | , (6.1) with unique initial and final mode vectors indexed e , randomly drawn from FE{ψi, φf ; τ} . 6.1 Experimental Assembly Complete quantum experiments Ee constitute an experimental sequence. Definition 6.1. An experimental sequence SeqEe represents experiments Ee on the system-under-study. All experiments in SeqEe are distinguished and order relevant. Classical experiments constitute a set SetEe , representing indistinguished and order irrelevant experiments. We see the same Moon, regardless of whether we repeat experiments or reverse their order. Quantum experiments are distinguished and order relevant. Nevertheless, repeating experiments on same photons in Malus experiment gives different results, but changing their order does not. Malus experiments constitute a series SerEe with experiments distinguished, but order irrelevant. Still more fundamental quantum experiments might constitute a sequence SeqEe . 6.2 Bimodal Mathematical Expectation For a complete individual experimental "trial" on the system-under-study, mathematical expectation (ME) of a predicate P of the system-under-study is given by P def= ME {P} = Tr (Pπ) Tr (π) = 〈φfin|P|ψin〉 〈φfin|ψin〉 , (6.2) where P is Schrödinger operator of predicate P ; ME : P −→ P . Here Tr (π) < 1 , except for parallel polarizer and analyzer. Definition 6.2. For an assembly of experimental "trials" on the system-under-study represented by SeqEe, experimental average of a predicate P of the system-understudy measured over N trials is given by Avg.P def= 1 N N∑ e MEe {P} = 1 N N∑ e Tr (Pπe) Tr (πe) = 1 N N∑ e 〈φfine |P|ψine 〉 〈φ fine |ψine 〉 , (6.3) where e is the index of trials. For N →∞ , (6.3) gives maximal informative average. ( xii - ) Bimodal Quantum Theory 7 Experimental Phase 7 Experimental Phase Initial and final statistical operators ρin = |ψin〉〈ψin| and ρfin = |φfin〉〈φfin| collapse initial and final "phases" exp (iθin) and exp (iθfin) of initial and final mode vectors ψin and φfin . Bimodal statistical operator π = |ψin〉〈φfin| reserves "bimodal phase" exp {i(θin − θfin)} , here termed "experimental phase". ρin and ρfin represent merely transition amplitudes, with no information of intact initial and final mode vectors. π represents transition amplitude as well as initial and final phases, collectively exp {i(θin − θfin)} . Unimodal statistical operators ρin and ρfin represent partial quantum experiments, with no information about experimental phase. Experimental phase exp {i(θin − θfin)} represents a complete quantum experiment; so does the bimodal statistical operator π = |ψin〉〈φfin| , that preserves it. For Malus experiment, experimental phase is exp {−i∆θ} . 8 Wave Mechanics and Objectivity 8.1 Objects and Frame Relativity Classical objects are absolutisitc; all experimenters agree upon their being. Quantum objects (if ever conceived by means of measuring complementary variables simultaneously) are relative; each experimenter creates his own object, that others do not agree upon. Moon is a classical object; all experimenters rely on its being. Quantum objects are yet hypothetical, but differ from classical ones in this semantics. If a hidden-variable theorist theorizes an object proper to each experimenter, he does not recede to classical causality as he plainly believes. Hidden-variable theories do not seem to be counter proposal to Copenhagen theories, even though they ever succeed. Quantum objects are possible, but relative (proper to an experimenter E). Classical objects are inevitable, and absolutisitc (regardless of experimenter). Quantum experimenter creates or invents quantum objects proper to his frame of experiment FE{ψi, φf ; τ} alone. He lacks transformation of such objects to other ones. Quantum objects are not invariant under quantum transformations; experimenters lack consensus for their being. Quantum "objects" appear for each experimenter's frame alone, like "time" in special relativity. Quantum theory is frame relativity or quantum relativity. Dirac called his frame relativity "Transformation Theory" [16]. Special relativity has no absolute "time"; experimenters or their frames have "proper time", but they lack consensus for it. Each experimenter has his own proper time. Quantum relativity has no absolute "object"; experimenters have "proper or apparent object", yet each has his own. They lack consensus for its being. Quantum theory relativizes "states-of-being", as Galilean relativity relativized "space" and special relativity relativized "time". Observers in special relativity have "unimodal frames"; experimenters in quantum theory have "bimodal frames". Special relativity relativizes "time", but retains composite "space-time" as reminiscent absolute [17]. Quantum theory relativizes "objects" and "states-of-being", but retains the "experimenter" himself as sole absolute. Bohr called relativization of experimenter "painful renunciation" [18], and retained classical concepts for the experimenter alone [4]. Some people misassimilate it as though he retained ontology for the system-under-study, and endow realistic in- ( xiii - ) 8.2 States of Being S. Dwivedi terpretations of ψ to Copenhagen theories. Quantum relativity has deeper semantics than special or general relativity [2] Quantum objects do not have semantic resemblance with classical objects. I doubt that quantum theory fully corresponds to classical in all logical, syntactic and semantic manner as }→ 0. Correspondence principle does not seem to respect logical, syntactic and semantic resemblances all together. Current form of quantum theory seems to have syntactic resemblance alone with classical theory. Our goal is to develop a radical syntax for the theory, that does not respect correspondence. 8.2 States of Being States (of being) pervaded physics since René Descartes, who, emulating Pythagoras and Plato, entailed equivalence between mathematical and physical structures. He eliminated role of observer 'I' and 'God' from the discussion of 'Matter' [18]; his split God|World|I. Cartesian objects were "vortices of plenum" that pervaded entire Cosmos. Cartesian world was envisaged an Object of objects. Newton brought empiricism to cartesian project, but retained dogmatic states-of-being. States are unimodal mathematical structures; mathematical idealistic objects that do not change in perception. We know them as they are. This dogmatic epistemology ruined physics until advent of quantum theory of Heisenberg (and Bohr), who sooner renounced it together with states-of-being [18]. Copenhagen division is God|World-I, where 'God' is split from composite 'World–I' (or entangled 'system–experimenter'). Quantum experimenter has no mathematical structure to represent state-of-being of the system-under-study. His own choices of (initial and final) mode vectors ψin and φfin constitute his frame of experiment FE{ψi, φf ; τ} , being unique to each experimenter alone [2]. Cartesian experimenters had a frame too; their assertions about the system-understudy were communicable to others. Cartesian experiments were idealistic; their transformation (or translation of assertions) entailed absolute objects, things in their own essence. Cartesian experimenters had a consensus for system's being (in a state); quantum experimenters have none. Quantum experimenters lack this communication or 'frame transformation'. They lack a dictionary to translate their assertions [2]. A quantum experiment changes system-under-study abruptly and irreversibly, leaving no trace for it to be correlated with subsequent experiments. Quantum systems "jump" from one experiment to another (or one mode to another), lacking causality. This jump is often called collapse or reduction of the state in hodge-podge classical-quantum language. Quantum theory renounces states-of-being together with possibility for these redundant notions. On the contrary, classical systems evolve from one experiment to another, carrying the germ of state-of-being. Blochinzev and Alexandrov objectivized ψ and ascribed it "state of being" of the system-under-study [18]. Wigner took it too far, and completely deviated from the Copenhagen doctrine [2]. His theory, often called "Orthodox theory", is completely dogmatic, where ψ, being statefuntion of the system, is an objective reality. Wigner ascribes consciousness to the abrupt change in ψ during measurement, and calls it a breakdown of quantum theory. Finkelstein (1991) calls the oxymora "statefuntion ψ of the system" syntactic error in the theory [19]. Ludwig (2006) calls it "Fairy Tale" ( xiv - ) Bimodal Quantum Theory 9 Correspondence or "Myth" [20], that plagues quantum theories of today. States are incompatible with bimodal quantum language. Finkelstein (1996) develops an algebraic "operational" language to study some prototypical "bimodal" quantum experiments [2], and develops its semantics. 9 Correspondence Quantum logic is bimodal. We see quantum systems in initial and final modes of experiment; intervention lies outside the theory. If one ignores bimodality and adheres to unimodal logic, he entails correspondence Qin → Qfin , and discrete triangle of Section 2.1.1 collapses to continuous line (or channel) connecting E with Q . Such a unimodal (no longer vertex) point Q represents unimodal state-of-being of the system-under-study. In unimodal quantum theories, ψin represents action vector connecting experimenter E with unimodal system-under-study Q . Specification of Q (with ω = 1) renders the system-under-study an "object". For an apparent system-under-study (with ω = 1), Q represents phase space of a classical system with coordinates (q, p) . Heisenberg [9] set out a limitation for their simultaneous measurement ∆q∆p ≥ }/2 , and q and p were called complementary variables in Copenhagen theory. Some adduce that Q indeed represents partially the state-of-being of systemunder-study, and as } → 0, Q represents it fully. Quantum theory corresponds to classical in the limit } → 0. Quantum theory was conceived an evolution of classical theory, and recedes to it as } → 0. It created an "interpretational problem" for quantum "measurement"; quantum theory became a problem itself, rather than a solution [2]. Such absurdity arises from the hodge-podge classical-quantum language. Quantum theory is based on bimodal logic, and has no logical counterpart in unimodal classical theories. Quantum experiments are pragmatic and bimodal. Classical experiments are dogmatic and unimodal. Quantum experiments are incompatible with classical language. Quantum language is subjective, bimodal and verb mode (Bohm's rheomode [14], Heisenberg's pragmatic [18] or Finkelstein's praxic [2]), that assimilates bimodal actions. Some people misinterpret subjectivity with experimenter's consciousness, and attribute it to experimenter's will. In Copenhagen theory, subjectivity is restricted merely to non-objectivity. Classical language is objective, unimodal and noun mode (dogmatic or ontic), that assimilates ultimate reality. They contradict each other in their usage, and one scarcely finds a mutual correspondence. Quantum experimenter puts "name" to his actions on the system-under-study. Classical experimenter puts "name" to the system-under-study itself, like "Moon". Quantum theory does not correspond to classical theory. Some variants of quantum theory correspond to classical theory in the limit } → 0. This correspondence is mere syntactic. They do not seem to correspond in semantics, while they lack one. These theories are based on hodge-podge classical-quantum language that Weizsäcker [1] calls semantic inconsistency. These theories are based on quantum syntax but classical semantics. Some people call it interpretational problem. Their correspondence to classical theories is obvious. Copenhagen theory does not correspond to classical theory. ( xv - ) 10 Summary S. Dwivedi 10 Summary Unimodal statistical operators ρin = |ψin〉〈ψin| and ρfin = |φfin〉〈φfin| represent incomplete or partial quantum experiments, and lack modal semantics [2]. Bimodal statistical operator π = |ψin〉〈φfin| represents complete or full quantum experiments, and reserves modal semantics. Unimodal variants of quantum theory, modeled on ρin = |ψin〉〈ψin| , are based on hodge-podge classical-quantum language, that Finkelstein (1972) calls "hybrid" [cq] theories [21]. Classical theories are [c], Copenhagen theories are [q]. The protocol of progress is [c] → [cq] → [q]. Ontic or causal quantum theories are [cq] , that correspond to [c] in the limit } → 0 ; [q] theories do not. Hidden variable theories seem to reduce [cq] and [q] theories to [c] ; [cq] theories are more likely to reduce to [c] , than [q] theories. ρin = |ψin〉〈ψin| and ρfin = |φfin〉〈φfin| represent [cq] experiments; π = |ψin〉〈φfin| represents [q] experiments, and resembles core insights of the Copenhagen theory. Copenhagen theory jumps from realm of classical physics, and creates the realm of quantum physics. Some people claim that Copenhagen theory has no proper form, and could not be assimilated due to lingual ambiguities alone. Heisenberg agreed at this point with Stapp [22]. Stapp adduces that Copenhagen theory is pragmatic. Finkelstein studies its modal semantics [2]. Copenhagen theory differs from other variants of quantum theory in its modal structure. Some theories, being operational, seem to be closer to Copenhagen theory. Theories that incorporate dogmatic states-of-being are unimodal. Some variants of quantum theory are pragmatic but unimodal. They are based on hodge-podge classical-quantum language, that increases ambiguity. These theories fit with experimental inferences, but lack a consistent semantics. Some people call it "interpretational problem", and many endure in vain to find one. Great many alternative interpretations of quantum theory have been published, and probably none discuss the original one. Some discuss ontological "causal" interpretation of quantum theory [23] (this school is in progress today), and others eliminate "observer" from the discussion [24, 25] as Descartes did. Copenhagen theory has been scarcely discussed after 1970's [2] and dogmatic "states-of-being" seem to dominate physics again, that we renounced in 1920's. Other variants of quantum theory are modeled on states-of-being in quantum domain, and discuss measurement problem in their interpretation. Measurement is the central problem of these theories, and has been widely discussed [26]. It gives rise to redundant concepts like "collapse" and "reduction" of wavefunction and state; quantum systems have no state to collapse [27]. Copenhagen theory is modeled on experiments, and has no measurement problem. Copenhagen theory is both pragmatic and bimodal, and owes no semantic error. Quantum theory has no interpretational problem. Bohr's insistence for classical concepts was restricted merely to the experimenter alone. He theorizes an absolute experimenter, and calls its relativization "painful renunciation". A theory of relative experimenters is in progress [4]. Some people misinterpret it as though Copenhagen theory insists classical concepts in dogmatic and causal sense, and endow states-of-being and realistic interpretations of ψ to Copenhagen theory. This work aims at assimilating core insights of the Copenhagen theory. ( xvi - ) Bimodal Quantum Theory REFERENCES Acknowledgements Work is not a new interpretation of quantum theory, yet an insight for the actual one. In the intervening moment, since its advent, many theorists seem to have deviated from what quantum theory is about, including author's earliest endurance. In apology to it, this work resembles core insight of Copenhagen theory. I owe this revelation to seminal writings of Werner Heisenberg, David Bohm and David Finkelstein. This work emerges from a "Talk" given at Max-Planck-Institut für Mathematik in Die Wissenschaften, Leipzig. I am indebted to Larry Horwitz, Matej Pavsic and Detlef Dürr for support and encouragements on various occasions. 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