Chapter Five

Heisenberg’s Uncertainty Principle and Copenhagen Interpretation

5.1 The Uncertainty Principle of Heisenberg (1927).

Heisenberg Uncertainty Principle states that no two conjugate properties of a particle (like position and momentum, Energy and time etc.) can be measured with accuracy greater than value of Planck’s constant. This Principle has been shown to be a built in reality of the Quantum Model. If we try to pin point the position, the momentum of the particle will be heavily uncertain. This is because they are correlated. The reason for this uncertainty is that the state of the particle is so changed during the process of the measurement, that we will have changed its conjugate element in that process. The implication of this statement is that in the phase space of displacement and momentum minimum size of the pixel is of the order of h. Thus anything within that range cannot be defined properly. The same goes to energy and time.

Dp Dq ³ h/4p

DE Dt ³ h/4p

Cartoon by John Richardson
for Physics World, March 1998
http://www.aip.org/history/heisenberg/p08c.htm

The full expression for any two observables A and B is given by

A and B are two observables,

and their corresponding operators,

denotes the commutator of and = -

denotes averaging for the state and

ΔX is the Standard Deviation of X given by .

5.2 Conjugate Observables

In general not all the classical physical observables of a system can be simultaneously known with unlimited precision, even in principle. There are several pairs of observables, which give different, but nonetheless maximal possible descriptions of the system.  These sets are sets are known as "maximal sets of commuting observables."  Observables from different sets are "noncommuting observables". These pairs can be normally known with precision and will depend only on our measuring tool.

In classical physics the errors of the observation are still there. As a result one would get a probability distribution for the initial values of the conjugate variables and something very similar to the probability function in quantum mechanics is still obtained.

The disturbing factor is that the state of the system does not have a specific value until a measurement is made. Does it mean that physical world has no independent existence apart from the observer? If so the two grand foundations of science are at stake – realism and determinism. Realism is the statement that the external world exist independent of whether someone is there to see or not. Determinism is the statement that there is a law – whether we know it or not - that links cause to effect in the physical world. If this is known we can predict the future from the present.

This led to the development of what is known as the Hidden Variable Theory. It assumes that if we had known the whole truth we could retain both realism and predictability. What is lacking in quantum theory is that we have only a part of the information. It is the hidden variable in our knowledge that causes the crunch. John von Neuman apparently proved through a slight twist in the mathematical formalism that no such theories could be developed. (This was later proved to be a false alarm based on a mathematical error by Bell)

So Bohr put forward his complementarity principle, which led to the Copenhagen interpretation of quantum theory.

5.3 The complementarity concept of Bohr

This proposes that the state of the system is the complete only with the microscopic system plus the macroscopic measurement apparatus. Then the complementary nature of wave-particle duality, and the character of the uncertainty principle are an intrinsic property of the nature and not a peculiarity of the measurement process.

In Bohr's Copenhagen interpretation, it does not make any sense to speak about characteristics of the quantum system in itself explicitly without the rest of the wider system including the measuring devices. It is therefore even more senseless to assign to a quantum system simultaneously complementary attributes like position or momentum, since the apparatuses, which are needed to measure them, are mutually exclusive. For example, it is impossible to build an apparatus, which simultaneously measures both position and momentum to arbitrary precision. So any question related to them cannot be asked intelligently. This interpretation received its crucial test at the Solvay Conference in 1927 in Brussels. Especially Einstein again and again discussed those experiments, which had always led to the worst paradoxes, in all details. New mental experiments were invented to trace any possible inconsistency of the theory. It was then seen that the theory was consistent as far as we can think of and seemed to fit all the experiments known so far.

[See Physics and Philosophy, Werner Heisenberg, 1958; Published by George Allen and Unwin Edition, 1959.]

5.4 Heisenberg identifies the state vector with the "knowledge of the system".

The two Einsteinian concepts of locality and seperability are quantum mechanically untenable. Locality principle states that measurements which are made far apart (at spacelike distances) cannot affect each other. Separability principle states that even when two systems, which have previously interacted, are separated , the measurements on any one of them should not make a difference in the other.

Quantum Mechanically the system by definition should include itself and also the rest of the cosmos, which interact with it. You cannot isolate a system nor separate it. This concept is then used to explain the collapse of the state vector to eliminate simple nonlocality problems. Thus we do not have an isolated particle or isolated person or even an isolated God. While there is a difference in terms of space and time and quality, all form together into a complete unity. This leads us to the concept of God in the Eastern Churches. While beings in existence are different from God himself, beings have their being within Him.

Act 17:28 for in him we live, and move, and have our being; as certain even of your own poets have said, For we are also his offspring.

5.5 The Minimal Interpretation

In quantum mechanics a system is supposed to be described by its quantum state, sometimes also called as its state vector. Given the state vector, one can derive probability distributions for all the physical quantities pertaining to the system such as its position, momentum, angular momentum, energy, etc. using their corresponding operators and evaluating

or <| A|>

The operational meaning of these probability distributions is that they correspond to the distribution of the values obtained for these quantities in a long series of repetitions of the measurement. If we make large number of measurements for any A on similar systems, the result will be portrayed by the probability distribution. This is true of any observable. Thus the explanation do not assume simultaneous measurements at all. Thus the uncertainty relations are simply statements about the spreads of the probability distributions of the several physical quantities arising from the same state. Thus in the minimal interpretation the uncertainty as limits on the simultaneous measurements as proposed by Heisenberg or Bohr, need not be supposed. In this case the uncertainty of measurement is inherent in all measurements. We cannot know anything with certainty as long as these properties are interrelated in the state. This brings to Bohr’s warned that the principle of complementarity not only "set a limit to the extent of the information obtainable by measurement, but also set a limit to the meaning which we may attribute to such information."

5.6 Distinct yet forms Part.

The whole creation is part of God. It is the assertion that we are distinct and isolated that leads to selfishness and egocentric sin. We cannot know an individual without knowing his circumstances, his relation with God and God’s relation with him. Redemption is when we partake of the divine nature and be in the likeness of His Son. This theology leads to the concepts of theosis or divinization. The simple statement of this is, that since creation and beings are created by God and since apart from God nothing existed nor can exist, it implies that creation is simply a part of God – not as an identity but as a part. Just as hands and feet and eyes are part of human body, we form a part of the body of God. Hence in order to know God or man we need to know the whole. It is a union with God in his Energies or union by grace making us partake of his divine nature, without our essence becoming one with the essence of God.

In this sense the rest of the cosmos and its relation to the particle creates the particle or the state of the particle. The very concept of the state of the system is defined only in this way. While the state contains all information related to the system, we could get the particular information by applying particular operators. Thus in order to know how God deals with man, we need to look into history. Each event in history provides a part of his character with all the uncertainties that are attached to the event.

5.7 Time Evolution

“In the sharp formulation of the law of causality-- "if we know the present exactly, we can calculate the future"-it is not the conclusion that is wrong but the premise.”

--Heisenberg, in uncertainty principle paper, 1927

In the Copenhagen interpretation, the wave function is only our way of representing that part of our knowledge about the history of a system, which is needed for calculating future probabilities for specific measurement results. . To give an example - in the case of the double-slit-experiment, the wave function makes it possible for us to calculate both the probability of finding the electron passing through a given slit, and the probability where it will strike on the screen. So it is especially impossible in principle to predict with certainty both through which slit the electron will go and where it will fall on the screen.

The notion of randomness is a perennial problem. To what extent can we predict the future? Is out limitation of knowledge that determines our inability to predict future? In the early 19th century Physics known today as Classical Physics was based on deterministic laws of Isaac Newton. Thus knowing the present state of the Universe, one could predict the future exactly. This is because the laws of nature are immutable and eternal. Predictability is fundamental to Science. We do use Statistics where such randomness is assumed as inherent in the system to predict the evolution in time. We have done well in that direction. Within limits of errors statisticians were doing very well.

Then quantum mechanics came along and it changed our view. Quantum Mechanics introduced probability and randomness at the basic root level of Physical laws. It is so disturbing that even Albert Einstein considered it as basically untenable and that it will destroy the foundations of Science. The modern study of nonlinear dynamics - The Theory of Chaos - revealed that randomness and unpredictability are indeed fundamental even in the Classical Physics. Things got even worse when it turned up even in Mathematics. G. J. Chaitin of IBM shows that there are theorems connected with number theory that cannot be proved because when we ask the appropriate questions, we obtain results that are equivalent to the random toss of a coin.

“God not only plays dice in physics but also in pure mathematics. Mathematical truth is sometimes nothing more than a perfect coin toss. “
Gregory Chaitin A Random Walk in Arithmetic

New Scientist 125, No. 1709 (24 March 1990), pp. 44-46 http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ns.html

In "classical physics,” the future motion of a particle could be exactly predicted, or "determined precisely,” from a knowledge of its present position and momentum and all of the forces acting upon it. Stated thus it is the basis of the law of Karma. It is perfectly logical. The problem is that one cannot know the precise position and momentum of a particle at a given instant. Therefore its future cannot be determined exactly. Since there is always indeterminacy in the measurement of the particle in the initial state we can only have a range of possibilities for the future motion of the particle.

The implications of this principle in soteriology are direct. We have a lot of conjugate measurements in the field. A few of them are: Sovereignty and Decrees of God; Freedom of Will of Man and Foreknowledge of God (Predestination); Inability and Responsibility etc. It is evident that if we insist on absolute measurement in one element the certainty of the conjugate element will correspondingly vanish. Hence there must be a phase space where there is uncertainty. We should conclude therefore that we would never be able to observe exactly both the conjugate elements simultaneously. However we should mention that these are not directly measurable observable quantities and the problem arise only when we look into the history (in the Bible) to see the actual working of these elements.

5.8 Sovereignty of God and Free Will of Man

Thus it seems that even God is bound by the Uncertainty Principle, and cannot know both the position, and the speed, of a particle. So God does play dice with the universe. All the evidence points to him being an inveterate gambler, who throws the dice on every possible occasion.
Does God Play Dice? Stephen Hawking

If we make a note of any Sovereign act of God in terms of the Freedom of will of man involved, we will certainly see an uncertainty. It is in this sense only that we could say both exist simultaneously. God in his sovereignty permits freedom of will within bounds thus limiting his own sovereign imposition. Man in his Freedom of will creates his own constraints and limits his own bounds and limits. The decrees of God are not independent of the behavior of Man or the behavior of Man independent of the decrees and laws of God. It also applies to the purposes of God and direction of history in terms of human freedom. God predestines in terms of his foreknowledge. This foreknowledge includes the freedom of will of man. Similarly God assigns and demands responsibility in terms of human ability. Within the phase space of these conjugate elements, there is a volume of uncertain space. This uncertainty says that we will never know the decrees of God absolutely; we will never know the predestined destinations absolutely. Put it in another way, God does not decree absolutely nor man’s free will is absolute. The latter is clearly evident, but we are reluctant to admit the former. At least in terms of the visible observable areas this is certainly true. I was in fact surprised when I came across a book called “God always has a Plan B”

5.9 Time Evolution, Determinism and Prophecy

The soteriological application to this approach will simply be that though the path taken by individual persons are not predictable and is totally dependant on the free will of the person; groups, the church, and mankind as a whole will be guided by the wavefunction directing the course of history to its predetermined culmination. So there is perfect freedom of will for each individual under constraints - which are bound to socio-psychic and genetic conditions. Yet history is perfectly guided to its predestined end. To put it directly, each individual is free totally within constraints, but God guides the path of mankind as a whole so that all eventually reach the predestined end according to the will of God for Mankind. This argument leads to the conclusions of Origen where all mankind will be eventually (not necessarily in this age or in the immediate ages to come. I certainly do not believe in reincarnation, which defeats the purpose. I believe in continuation of life and personality through resurrection) saved, but each one has the perfect freedom to take their own trajectory. They can even move back against God’s direction. It is the constraint that directs the envelope. God’s direction is like a carrier wave.

1 Tim 2: 3 This is good, and pleases God our Savior,4 who wants all men to be saved and to come to a knowledge of the truth.

2 Peter 3:9 The Lord is not slack concerning his promise, as some men count slackness; but is longsuffering to us-ward, not willing that any should perish, but that all should come to repentance.

John 1:9 That was the true Light, which lighteth every man that cometh into the world.

Most evangelicals find a threat to the gospel in such an interpretation. But is it there? We should be aware that the gospel of salvation to mankind, which we preach today, is limited to this age. We are not given any revelation regarding the ages to come, though we are positively guaranteed that they do exist. There indeed will be a separation of the sheep and the goats, the right and the left, one that goes to eternal heaven and others to eternal punishment. Eternal is an undefined mathematical concept indicating ignorance. But heaven and punishment are clearly defined. The message is clear that the time of salvation is now and the result of rejection is unimaginable.

While a particle is localized in a particular volume, its electron still has a finite though infinitesimal chance of being at the other end of the galaxy. What holds together the particle in place is the reality that it is only for an infinitesimal period of time? This permits a lot of opportunity for freedom of movement for the electron without losing the property of the particle as a whole. Human thinking and the freedom of man are similarly unbounded. What defines the person is the time average (of conduct), which determines the properties (character) of the person. The choices form a continuous chain of events. It is in the direction of these ongoing choices that we see as the personality of a person. This is no new teaching or experience. Even the very best of men err and even the worst of men do well occasionally.

While determinism is totally ousted, there is a probabilistic evolutionary concept. Can we predict future exactly?

The interesting thing about it is that though the trajectories of individual particles are not predictable the probabilities of each motion, and the distribution of many particles following these motions, could be calculated exactly from Schrödinger wave equation. How does this translate into soteriology? Even though we cannot predict the action path of each individual, it is still possible to predict the path of a group, a community, a church, or a nation with almost certainty. For an individual person we can still predict a probability distribution, which inherently defines uncertainty of exactness.

While the enveloping guiding wave moves history towards a predetermined end, the history of nations and communities and families and ultimately of individuals are not that clearly defined. This will allow predestinations in spite of freedom of will. There is always indeterminacy but it becomes smaller and smaller as the sample gets larger and larger.

Let me take a historical situation of Nineveh. Even though the prophet declared unequivocally without a condition that God was going to destroy Nineveh in forty days. God changed his mind. Because of the free will of the people of Nineveh, the prophecy did not come to pass. The impossible happened: the people of Nineveh believed God, the greatest of them to the least of them.

Jon 3:4 Jonah began to go into the city, going a day's journey. And he cried; "Yet forty days, and Nineveh shall be overthrown!"

Jon 3:5 And the people of Nineveh believed God; they proclaimed a fast, and put on sackcloth, from the greatest of them to the least of them.

Jon 3:10 When God saw what they did, how they turned from their evil way, God repented of the evil which he had said he would do to them; and he did not do it.

In most religious traditions God experience is always emphasized as a personal experience. While it is certainly a personal experience the indeterminacy principle makes each experience only a possible observable among many. The certainty can be attained only in collective experience or repeated experience. This is obviously the scientific approach. Thus while the God experience is intensely personal, the validity of the experience is tested only through collective consensus. Thus the reality of Yhvh is based not on Prophetic utterances or on individual experience but on Israel’s experience of the Crossing of the Red Sea, The appearance of Yhvh in Mount Sinai and in the Covenant Dinner when the 72 elders saw the God of Israel.

Exo 24:9 Then Moses and Aaron, Nadir, and Abi'hu, and seventy of the elders of Israel went up,

Exo 24:10 and they saw the God of Israel; and there was under his feet as it were a pavement of sapphire stone, like the very heaven for clearness.

Exo 24:11 And he did not lay his hand on the chief men of the people of Israel; they beheld God, and ate and drank.

All the rest of the revelations got their validity from the Mosaic revelation for which over a million witnesses were there to vouch. The same is true of the Christian experience. Christ experience is not only based on individual transformation experience of the believer, but is based essentially and validated by the historical revelation of the resurrected Jesus for which we have massive witness and massive evidence.

1Co 15:17 If Christ has not been raised, your faith is futile.

Einstein never accepted Quantum Mechanics. His first objection was the probabilistic aspect of the time evolution operator. This seems to imply that the future is random. Einstein repeatedly said, "God does not play dice with the universe,” Bohr’s response was: "Quit telling God what to do!" In other words we cannot define God and use it as an axiom for further logical derivation. It should be the other way round. Our understanding of God is limited by how God reveals himself to us through history.

Personally I believe that the fact that we are left with a probabilistic aspect for any given observable is indicative of the reality that we are not really dealing with a materialistic world alone. This cosmos itself is only a subset of a wider set and our space-time is only a small part of the infinite dimensions of the reality. We can never describe the state with the observables within the real world because there is an unseen world beyond and the state vector exists in this multi-dimension.

This multidimensional aspect was even present in the elementary Bohr Theory of Hydrogen where the vector model was successfully employed. Angular momentum quantum numbers and the spin quantum numbers were obtained by projection of a vector along the magnetic field of reference.