The EPR Paper

decoherence

In 1935, three quantum physicists, Albert Einstein, Boris Podolsky, and Nathan Rosen wrote an article for the journal Physical Review, entitled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Quantum mechanics is the attempt to describe the physics of subatomic particles’ behavior and interactions using mathematics. Realism is the idea that objects, particles, and the like have existent properties in ontological reality, whether or not they are observed by a conscious being. Locality is the theory that objects are not able to affect each other instantaneously from a distance. In the EPR paper, the three physicists posited that it could not be the case that the current theory of quantum mechanics, realism, and locality are all simultaneously true. Two of any of the three, it was argued, could be compatible, but not all of them. In this paper, I will attempt to be concise with the mathematics, so as to be able to leave room to explain the overall philosophical picture of what the equations entail about reality. Also, Arthur Fine’s entry on the EPR paper in the Stanford Encyclopedia of Philosophy has been used to supplement my clarity but the conclusions drawn within come fully from the EPR paper and I have used no direct quotations from it.

The paper begins with a small preamble that states that a theory of physics can only be complete if that, for every element of reality being studied, there must be a corresponding part in the theory. This is referring to quantitative values of physical phenomenon, such as the momentum or spin of a particle. So, there must be a total explanation for each of these quantities within a theory that concerns their mechanism for it to be considered complete. Further, this must be done without “disturbing” the system (i.e., measuring it), and the theory must predict actual results with 100% accuracy (Einstein, et al. 777).

Next, the preface goes on to iterate that knowledge of one non-commutative operator prevents the knowledge of the other. A good, relevant example of this would be Heisenberg’s Uncertainty Principle, which states that the precise momentum and location of a particle cannot be simultaneously known. It is then stipulated that because of this, either the description of the wave-function (such as the Schrodinger Equation) must be incomplete, or that the two non-commutative quantities do not simultaneously exist in reality, which logically leads to taking a position against realism (Einstein, 777).

Next, the foreword states that if one attempts to make predictions about a system, using measurements of another system that has interacted with it, then both the wave-function is incomplete and these two quantities cannot have simultaneous reality, because the system is being altered by measurement. No wave-function contains counterparts for both position and momentum. In fact, the Copenhagen interpretation asserts that reality does not exist at the quantum level until actually measured, which this paper attempts to refute. So, the preface ends with the assertion that the reality described by the wave-function is incomplete. The paper wants to show that if quantum mechanics were complete, then these incompatible qualities could both have real, physically instantiated values (Einstein, 777).

After the short preamble, the introduction elaborates upon how, when considering a theory, one must keep in mind the distinction between objective reality, which is “independent of any theory,” and the physical concepts the theory uses to operate. For example, one could try to describe the relation between supply and demand using mathematics (or other methods), but they would have to keep in mind the objective reality of the relationship between the two, regardless of the constructs and variables the theory uses in its attempt at description.

Next, the paper goes on to stipulate that the “success” of a theory depends upon two conditions; that it is correct, and that it the “description is complete,” meaning that it takes into account all of the given factors in the system being described. The paper then states that for a theory to be correct, it must agree with human experience; in this case, whether it can make correct predictions. The human experience allows us to make inferences about reality, it says, which in the realm of physics are experimentation and measurement of results (Einstein, 777).

The next paragraph further reiterates the metaphysics of a definition of the completeness of a theory (both the ontology and the epistemology). It begins with the statement that for every part of the physical reality, there must be a counterpart in the physical theory; this is designated as the “condition of completeness.” The following paragraph states that parts of physical reality cannot be attained through a priori reasoning, but rather, through empirical, a posteriori investigation through experimentation and the subsequent measurement of the data. However, it is conceded that a full definition of reality is not needed. Instead, it only requires that if the value of a physical quantity can be guessed by the theory correctly all of the time, so then there must be a part of physical reality that corresponds to the value that results from the equation. The paper states that while this line of reasoning does not use up every possible way to describe physical reality, it still gives a way of determining the truth of a theory within the system that is being attempted to describe in the objective, real world. Indeed, the paper states that this is “regarded not as a necessary, but merely as a sufficient condition of reality” and that it is compatible both with classical mechanics and quantum mechanics in their view of reality (Einstein, 777).

Next, the paper lays some foundations for going into the formalism of quantum mechanics involved in the problem the paper is attempting to explain. It first goes into describing the behavior of a given particle with a single degree of freedom; a one-dimensional state in a system. State is then discussed, choosing the variable ψ to represent the wave-function, which is “a function of the variables chosen to describe the particle’s behavior.” Then, for each observable variable A, the same letter A is used to represent the operator of said variable. This is required by the prior stipulation that each quantity that can be observed in reality must also be used in the theory that is attempting to explain how the system works (Einstein, 778).

In the subsequent paragraph, it is explained that if ψ is an eigenfunction of A, that is, where ψ can scale the operator A without changing its direction and remaining nonzero, and where ψ’ corresponds to , where a is a number resulting from the operator A, then the physical quantity A must have the value a when in a state given by ψ. This means that if ψ is an eigenfunction of A then A equals a when in a state of ψ (Einstein, 778).

Next, the paper goes on to talk about how, given these conditions, that if there is a particle in the state ψ, then there must be a part of physical reality that corresponds to the quantity A. This makes sense, given the conditions previously required by the previously mentioned criterion of reality, where the value of a quantity in the theory can be predicted perfectly accurately, that it must correspond to actual quantities, so long as the system is not disturbed. The paper then goes on to provide a hypothetical function where ψ equals e to the power of (2 times pi times i over Planck’s constant) times (some constant number multiplied by an independent variable). From there, since that equation results in the operator that corresponds to the momentum of the particle of interest, another equation results that makes certain for us that the momentum of the particle mentioned in the ψ=e(2πi/h)p0x equation is an actual, real momentum (Einstein, 778).

The paper goes on to state that, given that the original conditions mentioned at the-beginning of the second-to-last paragraph, then it cannot be said that A has any particular value. The paper gives the example of the coordinate position of the particle being indeterminate. So, if the operator q is used as a multiplier of x (the independent variable) then we are given an equation that tells us that qψ=xψ but does not equal . Because of the theory of quantum mechanics, we can’t find the exact position of the particle between a and b, only a probability where it will lie, a derivative equation results that is equal to b minus a which entails that the particle could be at any given coordinate, which makes it impossible to determine the position of the particle (Einstein, 778).

Because of this, the paper goes on, the particle in the ψ=eet cetera equation determined earlier cannot be predicted and is thus only able to be found with direct measurement, but that violates the prior stipulation that the particle not be disturbed. So, the particle will no longer be in the state that the equation has determined. This is what leads us to believe that when you ascertain the momentum of a particle, you lose the ability to ascertain its location in space. Try as one might, there exists the impossibility to determine the position/momentum of a particle when the momentum/position is found (respectively), thus again, the knowledge of the first quantity is lost (Einstein, 778).

Because of this, the paper concludes from these premises that either the wave-function description of reality is incomplete, or the two quantities cannot coexist; this refutes either the completeness of quantum mechanics or realism. The values would be contained in and determinable by the equation if the wave-function was complete, which quantum mechanics takes for granted. The next part of the paper explains how the assumption in quantum mechanics of the completeness of the wave-function creates a contradiction with the criterion of reality (Einstein, 778).

The second part begins with the thought-experiment of allowing two systems, I and II, to interact for an amount of time 0 to T, where after T there is no more interaction. Also, it is assumed that the systems’ states before time 0 were known. Using Schrodinger’s equation, the combination of the states of systems I+II can be found for any time leading up to T. The combined wave-function can be designated as a variable (here, Ψ) in order to be set up for more formal equations. However, their individual wave-functions following the reaction cannot be (yet) determined, as this would require more measurement which requires reduction of the wave-packet (Einstein, 779).

Ψ is used as a function of x1, and then an infinite-series equation is induced, using the quantity A to pertain to system I with eigenvalues a1, a2, et cetera. This infinite series-involving equation determines x2, which denotes the wave-function of system II. The summation of the series reduces to a single, determinable variable through wave-packet reduction; the process of the aforementioned equation. This produces a wave-function for system II. Then, another similar equation is made using wave-packet reduction, but it uses B instead of A, so it comes up with a different expansion. This also gives us a wave-function for system II. So we end up with two separate wave-functions for system II; two different functions within one reality that are able to be calculated and must be occurring simultaneously following the second system’s interaction with the first (Einstein, 779).

Now it is assumed for sake of the math that these two wave-functions are eigenfunctions of two physical quantities, P and Q. Assuming systems I and II refer to two individual particles (as in the states of entangled electrons), another expansion is made, this one spanning the series of negative infinity to positive infinity. Using A as the momentum of the first particle, another equation can be made that will denote its eigenfunctions. This equation leads to a continuous spectrum so another series expansion ranging from negative infinity to positive infinity results. Using B instead of A results in an eigenfunction equation that contains Dirac’s delta-function. Once enough following equations are completed, we have found two different eigenfunctions of two noncommuting operators that correspond to physical quantities, the eigenfunctions of the operators P and Q (Einstein, 780).

From here, measuring A or B can lead us to be able to predict without measurement the value of P or Q. These both qualify as being consistent with the criterion of reality. However, two separate functions used in the preceding equations both correlate to two different wave-functions coexisting within the same, singular reality. The earlier-proven premise that either F) quantum mechanics, through the wave-function, is an incomplete description of reality, or G) two or more operators that do not commute cannot be simultaneously real, leads to a conclusion. As we have ruled out using mathematics that there can be two simultaneously correct wave-functions, the logical rule modus tollendo ponens (“MTP”) leads to a conclusion. The disjunction that F and/or G, and it is not that G, therefore it must be that F, makes it necessary that quantum mechanics’ understanding of the wave-function is incomplete (Einstein, 780).

However, the EPR paper admits that the disjunction of “F and/or G” can be disputed. Perhaps G could only be correct if the two wave-functions can be simultaneously predicted (or measured). Then the disjunction of F and the original formulation of G would not leave us able to rule out quantum mechanics’ ability to properly describe reality, as the disjunction would be requiring either one thing that may or may not be true, or another thing that is too liberal in its requisite to prove the other correct by its negation. More simply, that the disjunction is incorrect in assuming it must either be that quantum mechanics is incomplete or that two wave-functions cannot simultaneously correct. Maybe neither conclusion (F or the now-more constrained G) has been negated and we are thus unable to assert the opposing conclusion. This is where the paper ends, leaving an implicit opening to the idea that there is a “hidden variable” involved (Einstein, 780).

Now I will attempt to take the EPR paper and describe it in more illustrative terms. Specifically, I will try to show how QM and realism, when both true, must make the idea of locality incorrect, in the light of an experiment. David Bohm, theoretical physicist, describes two particles’ separation after being produced from some collection of energy. I will use electrons as an example. (Fine, 3.1) When these two electrons (one negatively charged, one positively charged, the positively charged one sometimes characterized as a “positron”) are at any distance, be they a foot apart or across hundreds of light-years apart, their entanglement destroys the ability for their spins to be simultaneously measured, as they have a shared wave-function. For example, one could try to measure the spin along the x-axis of one electron (thus resulting in knowledge of the other, opposite spin of the other entangled electron), and one could try to measure the spin of the other electron along the y-axis, but it wouldn’t work, thanks to Heisenberg’s Uncertainty Principle. The spin of the y-axis would ultimately be unmeasurable, because these two cannot be simultaneously known. Only one degree of freedom can be known for certain, and trying to determine a second axis of spin proves futile. This is illustrated quite well by the Stern-Gerlach Experiment. This prevention of measurement indicates that these electrons are somehow in communication with each other. The electrons would not only have to be in communication at a speed faster than light, which is impossible according to Einstein’s special relativity, but in fact, the communication would actually have to be instantaneous (Fine, 3.1). How could this be? How could two particles “know” anything, instantly, no less?

It must be that the particles perhaps somehow are determinately opposite at one degree of freedom immediately (or before) once shooting away from each other, then “know” independently to randomly change at the first once the attempt to be measured along the second is made, and/or have a nearly unimaginable ability to “talk” to each other, no matter the distance, instantly, in order to make second-degree measurement impossible, which seems even more insane. Both of these conditions seem to validate the results of the experiment. Yet these conditions violate special relativity and locality, so therefore, locality, QM, and realism cannot all be correct within the theses laid out in the EPR paper.

Works Cited

Einstein, Albert, et al. “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Review. Volume 47, Issue 10: pp. 777–780.

Fine, Arthur, “The Einstein-Podolsky-Rosen Argument in Quantum Theory.” The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), URL = <http://plato.stanford.edu/archives/win2012/entries/qt-epr/&gt;

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