Generalized Uncertainty Principle Saurav Dwivedi∗ December 12, 2010 Abstract Quantum theory brought an irreducible lawlessness in physics. This is accompanied by lack of specification of state of a system. We can not measure states even though they ever existed. We can measure only transition from one state into another. We deduce this lack of determination of state mathematically, and thus provide formalism for maximum precision of determination of mixed states. However, the results thus obtained show consistency with Heisenberg's uncertainty relations. keywords State • Praxism • Ontism • Schrödinger Operators • Probability • Uncertainty • Indeterminism • Entanglement PACS 03.65.Wj; 02.30.Tb; 02.50.Cw; 03.65.Ud Contents 1 Introduction i 2 Probability Eigenvalue Formalism ii 3 Quantum Dynamical Equations iv 4 Generalized Uncertainty Principle v 5 Ultimate Indeterminism vi 6 Conclusion vi A Unit Operator vii B Mode Vector vii 1 Introduction Our search is for a valid interpretation of quantum theory. Quantum theory does not seem to have form of a fundamental theory. For it does not involve "Law of Nature", but only law of the system under study [26, Finkelstein 2005]. It is rather a phenomenological theory [27]. "Quantum Mechanics" seems misnomer for a non-mechanistic theory. Quantum theory (and its semantics) is in contradiction with "mechanistic order" [6, 7, Bohm 1980, p. 173]. The mechanistic order conflicts with practical quantum theory in the sense of non-locality and entanglement. There is no mechanistic quantum theory. The basic idea (basis) of Mechanics is Ontism. The basis of Quantum is Praxism [23, Finkelstein 1996]. In this connection, the nomenclature "Quantum Mechanics" reflects classical ideology; vestige of classical thought, therefore. The irony is that classical ideology is complete, while quantum ideology is incomplete. ∗Department of Physics, Faculty of Science, Banaras Hindu University, Varanasi 221005 India. E-mail: Saurav.Dwivedi@gmail.com, Homepage: www.geocities.ws/dwivedi i 2 Probability Eigenvalue Formalism S. Dwivedi Quantum theory seems complete phenomenologically, for it can explain (all) phenomena (and experimental facts) in its domain. It is incomplete on ideological ground. This quantum spontaneity is subject to the very definition of "Reality" (as it is). Quantum theory is a theory of actuality, not reality. Actuality (paradox) is a scientific truth that can be proven scientifically, while reality is an "orthodox", illusion however. There is no room for an "absolute reality" or "absolute state" (of being) in quantum theory; as there is no room for "absolute time" in relativity. Physicists use a 'state' vector |ψ〉 εH in order to define 'state of a system' (as if it existed). A state vector actually reflects our actions upon the system, not any its properties [24]. What we measure is a Hermitian observable, not state itself [25]. The wavefunction ψ can not be constructed for a system. It is main underlying assertion of my approach. If a system existed (in isolation), having a probability of existence (w) [18, Dwivedi 2005], its wavefunction can not be constructed by means of properties of the system. Properties of a system (in classical sense) can completely be determined by means of Hamilton-Jacobi function 'Action' s(qi, t). It absorbs certain constants of integration a system has. It suffices system's complete determinacy. It was a successful formulation of classical ideology. Wavemechanics is not a successful formulation of quantum ideology. Not even self-consistent [25]. We have two (parallel) versions of quantum theory: - Praxic theory. Matrix-Mechanics invented by Heisenberg. The fundamental physical entities are actions, operations and processes. - Ontic theory. Wave-Mechanics invented by Schrödinger. The fundamental physical entities are states of being ; absolute reality. The Ontic theory better survived because it is more visualized. The Praxic theory has been overlooked [25]. Although the two versions are equivalent methodologically, Praxic theory is more valid than Ontic, ideologically. It is self-consistent too. The distinction state for ψ has been claimed absurd [4, 3, 23, 45, Weizsäcker 2006, p. 263]. In this paper I claim that Ontic theory assumes an state that can not be constructed. Quantum theory is a theory about probabilities and predictions [45, Weizsäcker 2006, p. 260]. However, I define system's existence in probabilistic manner. I assign a probability (w) in order to define a system in isolation. For w = 1 system is in pure state and all its variables are accessible, for w ε (0, 1) it is in mixed state as certain of variables are hidden or non-accessible (e.g. in presence of many type of interactions [35]). For w = 0 the system is in forbidden state and all its variables are hidden and system can be represented by none. I quantize this observable using Schrödinger's quantization rule and obtain ŵ = −i}∂/∂s . However, the two observables (s and w) are found incompatible and we obtain an uncertainty relation σsσw ≥ }/2 . As a consequence, for a pure state (σw = 0, σs =∞), s is undetermined and ψ can not be constructed. Thus if system existed, it can not be specified by a vector in Hilbert space. The state vector can not be constructed with certainty. After all I provide a maximum precision for construction of state vector for system in mixed state. 2 Probability Eigenvalue Formalism We have a general form of Schrödinger's wavefunction1 belonging to system's Hilbert space H, in generalized perspective [5] ψ(R(qi, t), s(qi, t)) := R(qi, t) exp ( i } s(qi, t) ) , i = 1, 2, 3, . . . , f , (2.1) which is orthonormalizable 〈ψα|ψβ〉 = ∫ +∞ −∞ ψ∗α(R(qi, t), s(qi, t))ψβ(R(qi, t), s(qi, t)) dτ = δαβ , (2.2) 1It is notable that ψ is function of qi and t implicitly as well as function of R and s explicitly. Although Action s[. . . ] is not function of qi and t necessarily, instead it is often functional of the path. It has been taken function of qi and t here for mere convention, that does not hurt assertion. Nevertheless, the result holds intact if one prefers ψ(R, s) := R exp ( i } s ) over (2.1). ( ii - ) Generalized Uncertainty Principle 2 Probability Eigenvalue Formalism where dτ (= f∏ i=1 hidqi , h being scale factor and f is degrees of freedom) is generalized volume element of the configuration space. [The system has all these variables, except ψ (and tacitly its space H) in Praxic perspective]. Differentiate (2.1) partially w.r.t. Action s(qi, t) to obtain ∂ψ(R(qi, t), s(qi, t)) ∂s(qi, t) = i } ψ(R(qi, t), s(qi, t)) . (2.3) I entail a unit (zero-order differential) operator that satisfies for an ordinary function f as well as for wavefunction (See Appendix A) If = f ; Iψ(R(qi, t), s(qi, t)) = ψ(R(qi, t), s(qi, t)) . (2.4) Following deduction (2.4) for (2.3), we obtain Iψ(R(qi, t), s(qi, t)) + i} ∂ψ(R(qi, t), s(qi, t)) ∂s(qi, t) = 0 , (2.5) which is in the form of eigenvalue equation. We deduce Schrödinger unit operator Î [in the sense of Schrödinger's quantization rule] satisfying unit eigenoperator equation [17, Dwivedi 2005] Î|ψ〉 = I|ψ〉 ; Î = −i} ∂ ∂s . (2.6) Its expectation value is given by inner-product 〈Î〉 = 〈ψ|Î|ψ〉 = ∫ +∞ −∞ ψ∗(R(qi, t), s(qi, t)) ( −i}∂ψ(R(qi, t), s(qi, t)) ∂s(qi, t) ) dτ = ∫ +∞ −∞ |ψ(R(qi, t), s(qi, t))|2 dτ = Prob. (−∞,+∞) . (2.7) [it could also be obtained alternatively using (2.4) and (2.6) in inner-product (2.7).] The operator Î, having trace Prob. (−∞,+∞), entails properties of our probability operator ŵ . For a system in isolation:  Prob. (−∞,+∞) = wpure = 1 for pure state; Prob. (−∞,+∞) = wmixed ε (0, 1) for mixed state; Prob. (−∞,+∞) = wforbidden = 0 for forbidden state. (2.8) Thus Î is essentially ŵ that satisfies probability eigenvalue equation ŵ|ψw〉 = w|ψw〉 ; ŵ = −i} ∂ ∂s . (2.9) Or wψw(R(qi, t), s(qi, t)) + i} ∂ψw(R(qi, t), s(qi, t)) ∂s(qi, t) = 0 , (2.10) having solution ψw(R(qi, t), s(qi, t)) = A exp ( i } ws(qi, t) ) . (2.11) For now we will treat ψ as function of s solely, for mere convention. For orthonormalization we have the inner-product, 〈ψw′ |ψw〉 = ∫ +∞ −∞ ψ∗w′(s)ψw(s) ds = |A|2 ∫ +∞ −∞ exp ( i } (w − w′)s ) ds = |A|22π}δ(w − w′) . (2.12) For A = 1/ √ 2π}, we have ψw(s) = 1√ 2π} exp ( i } ws ) (2.13) ( iii - ) 3 Quantum Dynamical Equations S. Dwivedi that follows Dirac orthonormality 〈ψw′ |ψw〉 = δ(w − w′) . (2.14) However, these eigenfunctions form complete set (ψ = ∑ w cwψw) . For (square-integrable) function ψ(s) ,2 ψ(s) = ∫ 1 0 c(w)ψw(s) dw = 1√ 2π} ∫ 1 0 c(w)exp ( i } ws ) dw . (2.15) The expansion coefficient is obtained by Fourier's trick 〈ψw′ |ψ〉 = ∫ 1 0 c(w)〈ψw′ |ψw〉 dw = ∫ 1 0 c(w)δ(w − w′) dw = c(w′) , (2.16) or c(w) = 〈ψw|ψ〉 . (2.17) Exploiting completeness (2.15), the amplitude R in (2.1) is obtained R = 1√ 2π} ∫ 1 0 c(w) exp ( i } s(w − 1) ) dw . (2.18) 3 Quantum Dynamical Equations Dynamics is a law relating physical quantities in course of time (or some internal observables [28]). In Praxic theory Action is a fundamental physical entity [23]. However, it could often be customary to deduce dynamics in course of Action. Let differentiate the inner-product, 〈Â〉 = 〈ψ|Â|ψ〉 = ∫ +∞ −∞ ψ∗Âψ dτ , (3.1) exactly w.r.t. Action with differential-integral rule fg(κ) = f ∫ +∞ −∞ φ(τ)K(κ, τ) dτ = ∫ +∞ −∞ f {φ(τ)K(κ, τ)} dτ , (3.2) we obtain (using chain rule for f := ∂∂s ) ∂ ∂s 〈Â〉 = 〈∂ψ ∂s |Â|ψ〉+ 〈ψ|∂Â ∂s |ψ〉+ 〈ψ|Â|∂ψ ∂s 〉 . (3.3) Considering probability eigenvalue equations |∂ψ ∂s 〉 = i } |ŵψ〉 , 〈∂ψ ∂s | = − i } 〈ŵ†ψ| , (3.4) we obtain ∂ ∂s 〈Â〉 = 〈∂Â ∂s 〉 − i } [〈ŵ†ψ|Â|ψ〉 − 〈ψ|Âŵ|ψ〉] . (3.5) Here A, defined by A = 〈ψ|Â|ψ〉, is a dynamical [28] - an observable-valued-function of system's variables - A(qi, t) as distinct from observables. Since probability is a real aspect of nature, i.e., in operator representation, it must be hermitian3, 〈ŵ†ψ|Â|ψ〉 = 〈ψ|ŵÂ|ψ〉 , (3.6) which yields ∂ ∂s 〈Â〉 = 〈∂Â ∂s 〉 − i } 〈[ŵ, Â]−〉 . (3.7) 2As Probability does not exist in the limit (−∞, 0) U (1,+∞) , we have omitted integration over this limit. It does not create trouble in formalism. 3It also follows from counter-intuitive behavior of probability operator ŵ. ( iv - ) Generalized Uncertainty Principle 4 Generalized Uncertainty Principle This is first order quantum dynamical equation. Following the analogy, we further obtain second and third order quantum dynamical equations ∂2 ∂s2 〈Â〉 = 〈∂ 2Â ∂s2 〉 − i } 〈{ [ŵ, ∂Â ∂s ]− + ∂ ∂s [ŵ, Â]− − i } [ŵ, [ŵ, Â]−]− }〉 , (3.8) and ∂3 ∂s3 〈Â〉 = 〈∂ 3Â ∂s3 〉 − 〈{ [ŵ, ∂2Â ∂s2 ]− + ∂ ∂s [ŵ, ∂Â ∂s ]− + ∂2 ∂s2 [ŵ, Â]− − i } ( [ŵ, [ŵ, ∂Â ∂s ]−]− + [ŵ, ∂ ∂s [ŵ, Â]−]− + ∂ ∂s [ŵ, [ŵ, Â]−]− − i } [ŵ, [ŵ, [ŵ, Â]−]−]− )}〉 . (3.9) For operators (∂ nÂ ∂sn , n = 0, 1, 2, ...) compatible with ŵ, these equations follow Ehrenfest's theorem ∂n〈Â〉 ∂sn = 〈∂ nÂ ∂sn 〉 . (3.10) It holds good for observables having simultaneous eigenstates with probability w. 4 Generalized Uncertainty Principle The hermitian observables s and w are incompatible as operators ŝ and ŵ do not commute. As a consequence of incompatibility, s and w can not have simultaneous (shared) eigenfunction. The eigenvalue equations are ŝψs = sψs and ŵψw = wψw provided ψs 6= ψw . Thus, s and w could not be measured simultaneously for a given eigenstate. However, there appears uncertainty between existence and measure of Action. The expectation value of their commutator is given by innerproduct 〈φ|[ŝ, ŵ]−φ〉 = i}〈φ|φ〉 , (4.1) for some arbitrary wavefunction φ := φ(s) . The commutator [ŝ, ŵ]− = i} (4.2) yields measure of uncertainty between s and w by virtue of generalized uncertainty principle [30] σ2sσ 2 w ≥ ( 〈[ŝ, ŵ]−〉 2i )2 , (4.3) or σsσw ≥ } 2 . (4.4) This is similar to Heisenberg's uncertainty relations [32, 33]{ ∆q∆p ≥ }/2 , ∆E∆t ≥ }/2 . (4.5) As consequence: for a pure state (w = 1, σw = 0) the spread (uncertainty) in measurement of s is σs =∞. However, for a pure state, Action s can not be measured with certainty, and wavefunction (2.1) can not be constructed out of it. It does hold for forbidden state too. On the contrary, for a mixed state (a poorly or statistically defined state) (w ε (0, 1), σw ε (0,∞)) Action can be measured with maximum precision limit ∆s∆w ≥ } 2 . (4.6) ( v - ) 5 Ultimate Indeterminism S. Dwivedi However, for a mixed state, wavefunction can be constructed with a maximum precision (4.6). It follows that in Ontic theory mixed state is more predictable than a pure state, despite its poor statistical definition. My uncertainty relation (4.6) is not contrary to Heisenberg's assertions for measurement of "momentum and position" or "energy and time". By inspection, if measurement of Action may be certain, then all dynamical quantities (energy, momentum,...) may be derived simultaneously from Action using Euler-Lagrange and Hamilton-Jacobi equations. However, it would destroy all uncertainty relations (4.5) (as quantities appear simultaneously) that contradicts usual quantum theory. Our assertion that Action can not be certainly (strictly) measured for a system followed by uncertainty limit (4.6) overcomes this contradiction. Action is a hidden variable for systems in pure state and eventually: ψ is hidden variable for systems in pure state. The generalized uncertainty relation (4.6) can be decisive to deal with problems underlying mixed state (as there is higher predictability than pure state). For example, quantum phase transition. In transition period, state of system is mixed and a (pre or pro) pure state is not certain (strictly defined) during transition. We can contemplate (4.4) to deal with quantum phase transitions. 5 Ultimate Indeterminism A quantum is neither a granule nor a wave. These features are its partial characteristics that show agreement with particular situations (experiments) when any of it is more apparent (dominant). On the contrary, we can still look for another peculiar kind of such feature, alien to physics. Wavemechanics elaborated equivalence between Hamilton's principle (particle, granule) and Fermat's principle (wave, ray) in a unified manner: matter-wave, and replaced the constant under study with planck's constant [43, Schrödinger 1933]. Nevertheless, the theory thus evolved was more visualized and explained phenomena adequately, but still uses classical ideas. A wave does not introduce a new kind of entity. It is based upon classical idea of a particle [in the sense of localization]. Thus wavemechanics is an Ontic theory that assumes absolute reality; state of being. The reason for its enormous success in quantum domain is that wavelength introduced by matter-wave hypothesis is of order of atomic diameter [43]. On the contrary, matrixmechanics was less visualized [34] but yet more valid ideologically. Praxic theory entails fundamental physical entity - action - which is quantized. A measurement exchanges quanta (nh) with the system. A subsequent measurement exchanges few more quanta and we are led to a new - perturbed - observation. Uncertainty arises from the theory itself, not from perturbation of measurement by observer [45, Weizsäcker 2006, ch. 9, p. 252]. A minimum uncertainty is indispensable because it appears due to quantization of action [34]. In classical domain discreteness goes over to smoothness and exchange of quanta during measurement vanishes with good approximation, and we are led to precisely same observation in repeated measurements. In quantum domain a minimum uncertainty is inevitable even though measurement were ultimately precise [Bohr]. Heisenberg suggested to exclude (minimum) uncertainty by merging object, observation, observer into one (whole) object-observation-observer [34, Heisenberg 1933]. It cries out for another relativization [23, Finkelstein 1996]. Nevertheless, I urge that (a minimum) uncertainty is indispensable and can only be excluded by means of changing postulates of the theory; not by relativizing it. Such a change would lead to yet another theory, and it remains that uncertainty can be eliminated from quantum theory. Einstein suggested hidden-variables at sub-quantum level that proceed more rapidly than quantum, in order to overcome with uncertainty, but no such theory has yet been developed satisfactorily [7, Bohm 1980, ch. 4]. 6 Conclusion In objective vision we can learn about a system without interacting with it. It is similar to paint a portrait in a dark room. In this context - objective reality - seems orthodoxical. We ought to ( vi - ) Generalized Uncertainty Principle A Unit Operator change the definition of reality, rather than fitting a theory with element of reality previously defined. Heisenberg called his quantum theory non-objective. In quantum domain measurement occurs via interaction with the system. The fundamental entity of interaction is a signal, which is quantized. A reality comes out to be what we observe, not what exists. In this connection, observation with observer itself is not separate from the system being observed. An observation depends upon the frame of experiment (explicitly on observer), not system's state of being. Epistemology changes. The fundamental physical entities are actions, operations and processes, not state of being. Instead of claiming incompleteness of the wavefunction [19], we emphasize that state can not be constructed out of it. Wavefunction is a catalogue of knowledge (probability) not objective reality [45]. The system does not has ψ whether it be interpreted as state vector [Almost majority], mode vector [23, Finkelstein 1996] or channel [4, Blatt and Weisskopf 1952]. Appendix A Unit Operator Unit operator (eigenoperator), analogous to identity matrix, is deduced as a zero-order (ordinary or partial) differential operator (irrespective of with respect to what) defined as I := ∂0x = ∂0 ∂x0 ; (x = q, p, t, ...) . (A.1) We have observed in mathematical analysis that a zero-order differential operator does not change the function to which it is applied which leads to deduce it unit operator satisfying If = f . For example, in Ostrogradsky transformation, zero-order prime of generalized co-ordinate (n) q , (n = 0, 1, 2, 3, ...) for n = 0 is given by q. It may be extended to (n) q = Iq = q for n = 0 with I := ∂0t . The deduction is less applicable in mathematical analysis but is very important to deal with quantum problems. Unit operator is quantized to Î := −i} ∂∂s satisfying unit eigenoperator equation Î|ψ〉 = I|ψ〉 while treating quantum problems. For example, a quantum transformation with (n) ψ , (n = 0, 1, 2, 3, ...) (being nth-order partial derivative of ψ w.r.t. any variable x) is extended for n = 0 , (n) ψ = Iψ = ψ with I := ∂0x . This is a quantum problem and we quantize I to Î which yields (n) ψ + i}∂ψ∂s = 0 , for n = 0 . B Mode Vector An state |ψ〉 ε H in Ontic theory is entailed definite for an operator Â, that transforms it into a hermitian observable α via linear equation Â|ψ〉 = α|ψ〉 . On the contrary, |ψ〉 does not transform operator B (non-commutative with Â; [Â, B]− 6= 0) into its trace β : B|ψ〉 6= β|ψ〉 . |ψ〉 is not a definite state for B . Instead of assigning a definite state to an operator, we entail channel (or mode of action) |ψ〉 that transforms both operators, being definite state of neither, in following manner: Â|ψ〉 = {α± σα}|ψ〉 ; B|ψ〉 = {β ± σβ}|ψ〉 , (B.1) satisfying σασβ ≥ 〈[Â, B]−〉 2i . (B.2) However, we exclude definite state with channel, (much as mode vectors excluded state vectors [23, 24, 25, 26, 27]) without changing postulates of the theory. 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