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On the Measurement Problem International Journal of Theoretical and Mathematical Physics 2014, 4(5): 202-219 DOI: 10.5923/j.ijtmp.20140405.04 On the Measurement Problem Raed M. Shaiia Department of Physics, Damascus University, Damascus, Syria Abstract In this paper, we will see that we can reformulate the purely classical probability theory, using similar language to the one used in quantum mechanics. This leads us to reformulate quantum mechanics itself using this different way of understanding probability theory, which in turn will yield a new interpretation of quantum mechanics. In this reformulation, we still prove the existence of none classical phenomena of quantum mechanics, such as quantum superposition, quantum entanglement, the uncertainty principle and the collapse of the wave packet. But, here, we provide a different interpretation of these phenomena of how it is used to be understood in quantum physics. The advantages of this formulation and interpretation are that it induces the same experimental results, solves the measurement problem, and reduces the number of axioms in quantum mechanics. Besides, it suggests that we can use new types of Q-bits which are more easily to manipulate. Keywords Measurement problem, Interpretation of quantum mechanics, Quantum computing, Quantum mechanics, Probability theory Measurement reduced the wave packet to a single wave and 1. Introduction a single momentum [3, 4]. The ensemble interpretation: This interpretation states Throughout this paper, Dirac's notation will be used. that the wave function does not apply to an individual In quantum mechanics, for example [1] in the experiment system – or for example, a single particle – but is an of measuring the component of the spin of an electron on abstract mathematical, statistical quantity that only applies the Z-axis, and using as a basis the eigenvectors of σˆ3 , the to an ensemble of similarly prepared systems or particles [5] state vector of the electron before the measurement is: [6]. The many-worlds interpretation: It asserts the objective ψα=ud + β (1) reality of the universal wavefunction and denies the actuality of wavefunction collapse. Many-worlds implies However, after the measurement, the state vector of the that all possible alternative histories and futures are real, electron will be either or , and we say that the u d each representing an actual "world" (or "universe") [7, 8]. state vector of the electron has collapsed [1]. The main Consistent histories: This interpretation of quantum problem of a measurement theory is to establish at what mechanics is based on a consistency criterion that then point of time this collapse takes place [2]. allows probabilities to be assigned to various alternative Some physicists interpret this to mean that the state histories of a system such that the probabilities for each vector is collapsed when the experimental result is history obey the rules of classical probability while being registered by an apparatus. But the composite system that is consistent with the Schrödinger equation. In contrast to constituted from such an apparatus and the electron has to some interpretations of quantum mechanics, particularly the be able to be described by a state vector. The question then Copenhagen interpretation, the framework does not include arises when will that state vector be collapsed [1, 2]? "wavefunction collapse" as a relevant description of any Many interpretations of quantum mechanics have been physical process, and emphasizes that measurement theory offered to deal with this problem, and here, we will list a is not a fundamental ingredient of quantum mechanics brief summary of some of them: [9, 10]. The Copenhagen Intrepretation: When a measurement of De Broglie–Bohm theory: In addition to a wavefunction the wave/particle is made, its wave function collapses. In on the space of all possible configurations, it also postulates the case of momentum for example, a wave packet is made an actual configuration that exists even when unobserved. of many waves each with its own momentum value. The evolution over time of the configuration (that is, of the positions of all particles or the configuration of all fields) is * Corresponding author: [email protected] (Raed M. Shaiia) defined by the wave function via a guiding equation. The Published online at http://journal.sapub.org/ijtmp evolution of the wave function over time is given by Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved Schrödinger's equation [11, 12]. International Journal of Theoretical and Mathematical Physics 2014, 4(5): 202-219 203 Relational quantum mechanics: it treats the state of a results and not both. That means that one of the elementary quantum system as being observer-dependent, that is, the events only will happen: either {}H or {}T , but not both state is the relation between the observer and the system at once [23]. [13, 14]. What if the state vectors were nothing but another Transactional interpretation: describes quantum representation of events in the sense of the usual classical interactions in terms of a standing wave formed by both probability theory? What if the state vector before retarded ("forward-in-time") waves, in addition to advanced measurement is just the representation of a sample space, ("backward-in-time") waves [15, 16]. and the result we get after measurement is just an ordinary Stochastic interpretation: it involves the assumption of elementary event, in a similar manner to the coin example, spacetime stochasticity, the idea that the small-scale and in this sense the measurement is an experiment in the structure of spacetime is undergoing both metric and sense of the word used in probability theory? If we could topological fluctuations (John Archibald Wheeler's reformulate classical probability theory in such a way that "quantum foam"), and that the averaged result of these allows the representation of ordinary events by vectors, then fluctuations recreates a more conventional-looking metric at this will lead to an entirely different understanding of the larger scales that can be described using classical physics, underlying mathematics of quantum mechanics, and hence along with an element of nonlocality that can be described to quantum mechanics itself. And this is the aim of this using quantum mechanics [17, 18]. paper. Von Neumann–Wigner interpretation: It is an interpretation of quantum mechanics in which consciousness is postulated to be necessary for the 2. An Alternative Way to Formulate completion of the process of quantum measurement Classical Probability Theory [19, 20]. The many minds interpretations: They are a class of “no In this section we focus on the reformulation of collapse” interpretations of quantum mechanics, which is probability theory, then we use this formulation in next considered to be a universal theory. This means that they section to reformulate quantum mechanics. We reformulate assert that all physical entities are governed by some classical probability theory in a similar language to that is version of quantum theory, and that the physical dynamics used in quantum mechanics. Later on, we show that this of any closed system (in particular, the entire universe) is formulation reduces the number of postulates used in governed entirely by some version, or generalization, of the quantum mechanics. Schrödinger equation [21]. First we will start by considering finite sample spaces. In this paper, we will trod a different route in trying to Here it will be presented an outline of the method to be solve the measurement problem. used in this formulation: Ω This subject of measurement in quantum mechanics was Having an experiment with a finite sample space , studied by many prominent scientists, including Heisenberg, There is always a finite dimensional Hilbert space H von Neumann, Wigner and van Kampen [22]. And more with a dimension equal to the number of the elementary events of the experiment. recent studies such as the one done by Theo Nieuwenhuizen (Institute of Physics, UvA) and his colleagues Armen Then, we can represent each event by a vector in H Allahverdyan (Yerevan Physics Institute) and Roger Balian using the following method: (IPhT, Saclay), found that altogether, nothing else than I the square of the norm of a vector representing an event standard quantum theory appears required for understanding is equal to the probability of the event. ideal measurements. The statistical formulation of quantum II Given an orthonormal basis of H , we represent each mechanics, though abstract and minimalist, is sufficient to elementary event by a vector parallel to one of these explain all relevant features. Since alternative basis vectors, such that no different elementary events interpretations involve unnecessary assumptions of one kind are represented by parallel vectors, and the square of or another [22]. the norm of the representing vector is equal to the In this paper, we will tread a different route in trying to probability of the elementary event. solve the measurement problem. And we will start by III Then every event is represented by the vector sum of contemplating, thoroughly, classical probability theory at the elementary events that constitute it. first. From (I) we see that the vector ϕ representing the When we toss a coin for one time, the sample space of impossible event must be the zero vector because: this simple experiment is [23]: 2 Ω={HT ,} (2) ϕϕ= ϕ =p() ϕ = 0 (3) Of course, as it is known, this does not mean that the coin So: has all of these possibilities at once, it is merely a statement ϕ = 0 (4) about the possible outcomes of the experiment. And after doing the experiment, we will get just one of these two And the vector representing the sample space must be 204 Raed M. Shaiia: On the Measurement Problem normalized, because: N ϕ =00 = ∑ ui (17) ΩΩ =p()1 Ω = ⇒ Ω=1 (5) i=1 Furthermore, we know that the probability of an event is As a result of (II) and (III) we see that Ω is represented equal to the sum of the probabilities of the elementary by: events that constitute it [23], for example if: N A={ ab , ,..., d } (6) Ω=∑ cuii (18) = So: i 1 And we see that: pA( )= pa ({ }) + pb ({ }) ++ ..
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