WHO KILLED SCHRŐDINGER’S CAT? QUANTUM FIELD THEORY DID

In 1935 Erwin Schrödinger described a hypothetical experiment to show that something is wrong with the conventional interpretation of Quantum Mechanics.

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The [wave-function] of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. [1]

KatzeSchrödinger’s cat soon became the most famous example of what is now called the measurement problem, “the most controversial problem in physics today”, [2] with more than 30 youtube videos devoted to it. (Less well-known is that Einstein proposed a similar bomb experiment to make the same point, saying “a sort of blend of not-yet and already-exploded systems [cannot be] a real state of affairs”.[3] )

The measurement problem arises because QM does not supply a picture of reality when no one is looking. Instead we have particles that are neither here nor there, states that are in superpositions, and equations that only give probabilities. Most physicists believe that these superpositions are real, and some even accept that the cat can be both half dead and half alive. Then there are physicists who prefer not to talk about reality.

I am a positivist who believes that physical theories are just mathematical models we construct, and that it is meaningless to ask if they correspond to reality, just whether they predict observations. – Stephen Hawking.[4]

Something was clearly missing.

That something came along later in the form of Quantum Field Theory — a theory that does provide a picture of reality, even when no one is looking. However there are various interpretations and understandings of Quantum Field Theory, while some physicists reject it completely. For example, N. David Mermin wrote in Physics Today, “I hope you will agree that you are not a continuous field of operators on an infinite-dimensional Hilbert space,[5] and Meinard Kuhlmann wrote in Scientific American, “quantum field theory… sounds like a theory of fields. Yet the fields supposedly described by the theory are not what physicists classically understand by the term field”.[6]

Among those who accept Quantum Field Theory, most follow Richard Feynman’s approach based on particles and virtual particles, while Julian Schwinger’s (and Sin-Itiro Tomonaga’s) version, which is based only on fields, is much less well-known.[7] Interestingly enough, Frank Wilczek reports that Feynman later changed his mind:

Feynman told me that when he realized that his theory of photons and electrons is mathematically equivalent to the usual theory, it crushed his deepest hopes… He gave up when, as he worked out the mathematics of his version of quantum electrodynamics, he found the fields, introduced for convenience, taking on a life of their own. He told me he lost confi­dence in his program of emptying space. [8]

While both approaches lead to the same equations, the physical pictures are very different. It is Schwinger’s Quantum Field Theory that we refer to in this article, but because this version is so little known, we must first give a brief description.

Definition of field. A field is a property of space. This concept was introduced by Michael Faraday in 1845 as an explanation for electric and magnetic forces.  However the idea that space has properties was not easy to accept, so when James Maxwell predicted the existence of EM waves in 1864, an ether was invented to carry the waves. It took many years before the ether was dispensed with and physicists accepted that space itself has properties:

The conception of an ether absolutely at rest is the most simple and most natural – at least if the ether is conceived to be not a substance but merely space endowed with certain physical properties. – P. Drude [9]

…space-time itself had become a dynamical medium – an ether, if there ever was one. – F. Wilczek [10]

Quantization. In 1900 Max Planck showed that the EM field is not continuous but is made of discrete units that he called quanta. In his Nobel speech Planck said, “Here was something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking”. (How right he was!) Later in 1922 the Stern-Gerlach experiment showed that a physical quantity (angular momentum) can have only discrete values. These results led to the use of Hilbert space as the mathematical way to describe quantities with discrete values.

In Hilbert space the possible values of a physical property are called eigenvalues and each eigenvalue has a corresponding eigenvector. The eigenvectors are mutually orthogonal. However the state of a physical property does not have to be an eigenstate.  It may be represented by an arbitrary vector with components from more than one eigenvector. For this reason one cannot talk about the value of a property, one can only talk about its expectation value. There are also operators in Hilbert space that cause the state vectors to change and are used to describe the dynamics of the physical property.

Quantized fields. In Quantum Field Theory, the continuum of field strengths is regarded as a limiting case of discrete values, with Hilbert space being extended to an infinite number of dimensions. (Remember, Hilbert space is a mathematical tool, not a real space.) The quantized fields so described are sometimes referred to as operator fields.

Quanta. Quanta are chunks of field that are spread out in space but act as units. Each quantum lives a life and dies a death of its own.  Quanta can exist in free or bound states.  Examples of free quanta are a photon emitted by a lamp or an electron emitted from a cathode.  Examples of bound quanta are the electrons around an atomic nucleus or the protons and neutrons in the nucleus.

Field equations. Finally, there are equations that describe the evolution of the fields. In the words of Frank Wilczek:

The move from a particle description to a field description will be especially fruitful if the fields obey simple equations, so that we can calculate the future values of fields from the values they have now… Maxwell’s theory of electromagnetism, general relativity, and quantum chromo­dynamics all have this property. Evidently, Nature has taken the opportunity to keep things relatively simple by using fields. [11]

QFT vs. QM. The difference between Quantum Mechanics and Quantum Field Theory is not so much in the equations. After all, the Schrödinger equation of QM is the non-relativistic limit of the QFT equation for matter fields. The difference is in their interpretation. While the Schrödinger equation describes the probability that a particle can be found at a given point, the QFT equation describes the field strength at that point. The QM wave-function for the electron in a hydrogen atom gives the probability of finding the electron at a point, but in QFT this “wave-function” gives the strength of the electron field at that point

Quantum collapse. The field equations, important as they are, do not tell the whole story; in fact, they don’t tell the most important part of the story: They do not describe energy transfer. Without the transfer of energy, nothing of significance can happen, e.g., detection. In QFT a quantum can only transfer its energy to an atom if the quantum is absorbed by (or “sucked into”) that atom while disappearing from all of space. We call this process quantum collapse. Examples of quantum collapse are a photon depositing its energy into a photoreceptor in the eye or a radiated quantum transferring its energy into an atom in a Geiger counter.

Quantum collapse in QFT is reminiscent of wave-function collapse in QM, but there is a big difference.  Wave-function collapse is a collapse of probabilities whereas in QFT the collapse is real. A field quantum literally disappears when its energy is transferred to another quantum. While quantum collapse may seem strange, there is nothing inconsistent it. Indeed, if the world is made of quantized fields, quantum collapse is inevitable. This is not the first time in physics history when something was known to happen without a theory to explain it.

Source theory. In his later years Julian Schwinger developed a theory that he called source theory, based on the important role that absorption and creation of quanta play:

We have spoken of particle creation, but equally important is particle detection [Note: particles is Schwinger’s term for “stable or quasi-stable excitations” in an underlying field] … In a general sense the particle is annihilated by the process of detecting it. The… processes used to detect a particle can be idealized as sinks wherein the particle’s properties are handed on… but sink and source are clearly different aspects of the same idealization, and we unite them under the general heading of “sources”. [12]

The fact that Schwinger’s source theory was never accepted caused him great disappointment. As he wrote in the preface to the above-cited book,

Developments,… in which the new viewpoint was most successfully applied, convinced me, if no one else, of the great advantages in mathematical simplicity and conceptual clarity that its use bestowed. The lack of appreciation of these facts by others was depressing, but understandable.

Non-locality. An aspect of quantum collapse that troubles some people is that it is non-local; that is, the field quantum disappears from all of space instantaneously. However this is necessary if quanta are to act as independent units. Besides, non-locality is an experimental fact and does not lead to any inconsistency. It may not be what we expected, but just as we accepted that the earth is round and that matter is made of atoms, so can we accept that quanta collapse.

Schrödinger’s cat. Now let us return to Schrodinger’s cat experiment as seen from the QFT perspective. In the first stage, the radiated quantum interacts with all other quanta that it encounters, as described by the field equations. These interactions are deterministic and reversible, and no energy is transferred. There are no superpositions; the state of the system is completely specified by the field strength at every point – or, more accurately, by the vector in Hilbert space that represents that field strength. This reversible state continues until the quantum collapses.

When the radiated quantum collapses into an atom, it disappears from all space and its energy is transferred to that atom. If the atom is located in the Geiger counter, it creates a Townsend discharge that trips the relay that releases the poison gas that kills the cat. Until then the cat is alive. After that the cat is dead. There are no superpositions.

Probability remains. Since quantum collapse is not described by the QFT equations, we are not able to predict when it happens. All we have are probabilities that are related to field strength, so we still don’t know the result until we look. But if we toss dice, or take a sock from a drawer blindfolded, we don’t know the result until we look. That doesn’t mean it didn’t happen.

Schrödinger’s dilemma. Schrödinger himself wanted to believe that the electron is a wave spread throughout all of space, but he had to face the particle behavior as seen, for example, in a cloud chamber.[13]

Quantum Field Theory

From the point of view of wave mechanics, the [particle picture] would be merely fictitious. I have, however, already mentioned that we have yet really observed such particle paths… We find it confoundedly difficult to interpret the traces we see as nothing more than narrow bundles of equally possible paths. [14]

The answer to Schrödinger’s dilemma is quantum collapse, but in this case it is not total collapse. The quanta (alpha particles) have too much energy to be absorbed by a single atom, so they transfer their energy in a series of collapses. As Art Hobson wrote:

The tracks are made by successive individual interactions between a matter field and gas or water molecules. The matter quantum collapses… each time it interacts with a molecule, while spreading out as a matter field between impacts. [15]

QFT not only offers a solution to the measurement problem, it resolves and explains many other difficulties. Here are just a few examples.

The Uncertainty Principle. Fields spread out in space. Heisenberg’s Uncertainty Principle is nothing more than Fourier’s theorem.

e = mc2. While the primary effect of the mass term in the field equations is to slow down the speed at which quanta propagate, it also causes the field to oscillate at a frequency f = mc2/h. On the other hand, the energy of a quantum is represented by oscillations in its field intensity according to Planck’s law e = hf. Combining these two equations gives e = mc2. Not only does this equation tumble right out of Quantum Field Theory, its meaning is seen physically in the oscillations that arise from both mass and energy.

Spooky action at a distance. Quantum collapse can also occur if two quanta are created together so that their properties are correlated. In this case if one of the two collapses, the other one must do the same, and it must do it instantaneously. Experiments with correlated photons have demonstrated that if the spin of one photon changes, the other spin also changes, no matter how far apart the photons are. (This is an example of internal collapse, as opposed to the spatial collapse described earlier.) Einstein called this “spooky action at a distance,” but collapse of two correlated quanta is no harder to accept (or spookier) than the collapse of one. If we can accept that a single quantum, spread over miles of space, can instantaneously collapse, it is not much of a stretch to accept that two correlated quanta can do the same.

The subatomic zoo. A calculation was recently performed to see if the field equations for quarks and gluons would predict the array of hadrons that has been discovered. First, masses of three of the newly-discovered hadrons were used to ascertain the basic properties of quarks and gluons. Energy was then added to the computer simulation to see if it would settle into stable concentrations. As Frank Wilczek tells the story:

With a sigh of relief, we note that… the calculated masses agree quite well with the observed values… Also remarkable is what you don’t see coming out of the computer… although the basic inputs to the calcula­tions are quarks and gluons, they don’t appear among the outputs! The Principle of Confinement, which seemed so weird and desperate, here appears as a footnote to complete and comprehensive reality-matching… Through difficult calculations of merci­less precision that call upon the full power of modern computer technology, [we have] shown that unbendable equations… account convincingly and in quantitative detail for the existence of protons and neutrons, and for their properties… I believe this is one of the greatest scientific achievements of all time. [16]

The paradoxes of relativity. The theory of Special Relativity, formulated by Einstein in 1905, is based on the postulate that the laws of physics are the same regardless of the state of motion of the observer, so long as it is uniform. From this principle follow effects that many people find difficult to accept. While many physicists prefer Einstein’s top-down approach, the bottom-up method based on fields explains why these strange things happen.

  • Things happen more slowly in a moving system because the interacting fields must travel a greater distance (despite the contraction).
  • Objects contract when moving because motion affects the interaction of fields that hold the object together. Space itself contracts because space is made of fields.
  • Nothing can go faster than light because everything is made of fields that propagate at a rate determined by the field equations.
  • Mass increases with speed because mass means resistance to acceleration and acceleration beyond the speed of light is impossible.

Conclusion. Quantum Field Theory is an elegant theory that rests on a firm mathematical foundation.  It supplies a simple answer to the measurement problem and it also resolves the paradoxes of Relativity and Quantum Mechanics that have confused so many people.  Some physicists have trouble accepting quantum collapse because it is non-local, but non-locality is a proven fact. Field quanta evolve in accordance with the deterministic field equations, and occasionally they collapse and deposit some or all of their energy into an absorbing atom. In some cases this collapse may lead to a macroscopic change, i.e., a “measurement”. What could be simpler? One can only wonder why this theory hasn’t been embraced and taught in all physics courses.  Maybe it’s time for physicists to WAKE UP AND SMELL THE QUANTIZED FIELDS.

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[1] E. Schrödinger, from a 1935 paper in Die Naturwissenschaften, trans. by John D. Trimmer in Proc. Am. Phil. Soc. 124, 5 (1980)
[2] www.informationphilosopher.com/solutions/experiments/wave-function_collapse/
[3] W. Isaacson, Einstein: His Life and Universe, Simon & Schuster, NY (2007), p. 456
[4] R. Penrose et al. The Large, the Small and the Human Mind, Cambridge University Press (1997) p. 169
[5] N.D. Mermin, Why Quark Rhymes With Pork, Cambridge University Press (2016), p. 210.
[6] M. Kuhlmann,“What Is Real”, Scientific American, August (2013), p.45.
[7] J. Schwinger, “The theory of quantized fields”: I. Phys. Rev. 82, 914 (1951), II. Phys. Rev. 91, 713, (1953), III Phys. Rev. 91, 728, (1953), IV. Phys. Rev. 92, 1283, (1953), V. Phys. Rev. 93, 615, (1954), VI. Phys. Rev. 94, 1362, (1954)
[8] F. Wilczek, The Lightness of Being: Mass, ether, and the unification of forces, (2008) p. 84, 89)
[9] A. Pais, Subtle is the Lord: The science and the life of Albert Einstein, Clarendon Press, Oxford, p. 121
[10] F. Wilczek, “The persistence of ether”, Physics Today (1999), p. 11
[11] F. Wilczek, The Lightness of Being: Mass, ether, and the unification of forces, (2008) , p. 86
[12] J. Schwinger, Particles, Sources, and Fields, Vol 1 (1989), p. 38.
[13] www.practicalphysics.org
[14] E. Schrödinger, Nobel lecture (1933)
[15] A. Hobson, Physics: Concepts and Connections , 4th ed., Pearson Prentice Hall (2007), p. 313
[16] F. Wilczek, op. cit. pp. 122-127