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Quantum mechanics: Myths and facts

Hrvoje NikolicTheoretical Physics Division, Rudjer Boskovic Institute, P.O.B. 180, HR-10002 Zagreb, Croatia.∗

(Dated: September 21, 2006)

A common understanding of quantum mechanics (QM) among students and practical users isoften plagued by a number of “myths”, that is, widely accepted claims on which there is notreally a general consensus among experts in foundations of QM. These myths include wave-particleduality, time-energy uncertainty relation, fundamental randomness, the absence of measurement-independent reality, locality of QM, nonlocality of QM, the existence of well-defined relativistic QM,the claims that quantum field theory (QFT) solves the problems of relativistic QM or that QFT is atheory of particles, as well as myths on black-hole entropy. The fact is that the existence of varioustheoretical and interpretational ambiguities underlying these myths does not yet allow us to acceptthem as proven facts. I review the main arguments and counterarguments lying behind these mythsand conclude that QM is still a not-yet-completely-understood theory open to further fundamentalresearch.

Contents

I. Introduction 2

II. In QM, there is a wave-particle duality 2A. Wave-particle duality as a myth 2B. Can wave-particle duality be taken seriously?3

III. In QM, there is a time-energy uncertainty relation3

A. The origin of a time-energy uncertainty relation3

B. The time-energy uncertainty relation is not fundamental3

IV. QM implies that nature is fundamentally random4

A. Fundamental randomness as a myth 4B. From analogy with classical statistical mechanics to the Bohmian interpretation5

C. Random or deterministic? 6

V. QM implies that there is no reality besides the measured reality6

A. QM as the ultimate scientific theory? 6B. From a classical variable to a quantumlike representation6

C. From the quantumlike representation to quantum variables7

D. From quantum variables to quantum measurements8

E. From quantum measurements to no-hidden-variable theorems9

F. From no-hidden-variable theorems to physical interpretations10

VI. QM is local/nonlocal 11

∗Electronic address: hrvoje@thphys.irb.hr

A. Formal locality of QM 11

B. (Non)locality and hidden variables 12

C. (Non)locality without hidden variables? 12

VII. There is a well-defined relativistic QM 13

A. Klein-Gordon equation and the problem of probabilistic in

B. Some attempts to solve the problem 14

VIII. Quantum field theory solves the problems of relativistic

A. Second quantization of particles 15

B. Quantum fields 15

C. Does QFT solve the problems of relativistic QM?16

IX. Quantum field theory is a theory of particles17

A. A first-quantized analog of particles in QFT17

B. Particles in perturbative QFT 18

C. Virtual particles? 19

D. Nonperturbative QFT 19

E. Particles and the choice of time 19

F. Particle creation by a classical field 21

G. Particles, fields, or something else? 22

X. Black-hole entropy is proportional to its surface22

A. Black-hole “entropy” in classical gravity 22

B. Black-hole “entropy” in semiclassical gravity23

C. Other approaches to black-hole entropy 24

XI. Discussion and conclusion 24

Acknowledgments 25

References 26

2

I. INTRODUCTION

On the technical level, quantum mechanics (QM) is aset of mathematically formulated prescriptions that servefor calculations of probabilities of different measurementoutcomes. The calculated probabilities agree with exper-iments. This is the fact! From a pragmatic point of view,this is also enough. Pragmatic physicists are interestedonly in these pragmatic aspects of QM, which is fine.Nevertheless, many physicists are not only interested inthe pragmatic aspects, but also want to understand na-ture on a deeper conceptual level. Besides, a deeper un-derstanding of nature on the conceptual level may alsoinduce a new development of pragmatic aspects. Thus,the conceptual understanding of physical phenomena isalso an important aspect of physics. Unfortunately, theconceptual issues turn out to be particularly difficult inthe most fundamental physical theory currently known –quantum theory.

Textbooks on QM usually emphasize the pragmatictechnical aspects, while the discussions of the conceptualissues are usually avoided or reduced to simple authori-tative claims without a detailed discussion. This causes acommon (but wrong!) impression among physicists thatall conceptual problems of QM are already solved or thatthe unsolved problems are not really physical (but rather“philosophical”). The purpose of the present paper is towarn students, teachers, and practitioners that some ofthe authoritative claims on conceptual aspects of QMthat they often heard or read may be actually wrong,that a certain number of serious physicists still copeswith these foundational aspects of QM, and that there isnot yet a general consensus among experts on answers tosome of the most fundamental questions. To emphasizethat some widely accepted authoritative claims on QMare not really proven, I refer to them as “myths”. In thepaper, I review the main facts that support these myths,but also explain why these facts do not really prove themyths and review the main alternatives. The paper isorganized such that each section is devoted to anothermyth, while the title of each section carries the basicclaim of the corresponding myth. (An exception is theconcluding section where I attempt to identify the com-mon origin of all these myths.) The sections are roughlyorganized from more elementary myths towards more ad-vanced ones, but they do not necessarily need to be readin that order. The style of presentation is adjusted toreaders who are already familiar with the technical as-pects of QM, but want to complete their knowledge witha better understanding of the conceptual issues. Never-theless, the paper is attempted to be very pedagogicaland readable by a wide nonexpert audience. However, tokeep the balance and readibility by a wide physics audi-ence, with a risk of making the paper less pedagogical,in some places I am forced to omit some technical details(especially in the most advanced sections, Secs. IX andX), keeping only those conceptual and technical detailsthat are essential for understanding why some myths are

believed to be true and why they may not be so. Read-ers interested in more technical details will find them inmore specialized cited references, many of which are ped-agogically oriented reviews.

II. IN QM, THERE IS A WAVE-PARTICLEDUALITY

A. Wave-particle duality as a myth

In introductory textbooks on QM, as well as in popu-lar texts on QM, a conceptually strange character of QMis often verbalized in terms of wave-particle duality. Ac-cording to this duality, fundamental microscopic objectssuch as electrons and photons are neither pure particlesnor pure waves, but both waves and particles. Or moreprecisely, in some conditions they behave as waves, whilein other conditions they behave as particles. However,in more advanced and technical textbooks on QM, thewave-particle duality is rarely mentioned. Instead, suchserious textbooks talk only about waves, i.e., wave func-tions ψ(x, t). The waves do not need to be plane wavesof the form ψ(x, t) = ei(kx−ωt), but, in general, may havean arbitrary dependence on x and t. At time t, the wavecan be said to behave as a particle if, at that time, thewave is localized around a single value of x. In the idealcase, if

ψ(x) =√

δ3(x − x′), (1)

then the position x of the particle has a definite value x′.The state (1) is the eigenstate of the position operator,with the eigenvalue x′. However, the position operatoris just one of many (actually, infinitely many) hermitianoperators in QM. Each hermitian operator correspondsto an observable, and it is widely accepted (which, as weshall see later, is also one of the myths) that the posi-tion operator does not enjoy any privileged role. Fromthat, widely accepted, point of view, there is nothingdual about QM; electrons and photons always behave aswaves, while a particlelike behavior corresponds only toa special case (1). In this sense, the wave-particle dualityis nothing but a myth.

But why then the wave-particle duality is so often men-tioned? One reason is philosophical; the word “duality”sounds very “deep” and “mysterious” from a philosoph-ical point of view, and some physicists obviously like it,despite the fact that a dual picture is not supported bythe usual technical formulation of QM. Another reasonis historical; in early days of QM, it was an experimentalfact that electrons and photons sometimes behave as par-ticles and sometimes as waves, so a dual interpretationwas perhaps natural at that time when quantum theorywas not yet well understood.

From above, one may conclude that the notion of“wave-particle duality” should be completely removedfrom a modern talk on QM. However, this is not nec-

3

essarily so. Such a concept may still make sense if in-terpreted in a significantly different way. One way ispurely linguistic; it is actually common to say that elec-trons and photons are “particles”, having in mind thatthe word “particle” has a very different meaning thanthe same word in classical physics. In this sense, elec-trons and photons are both “particles” (because we callthem so) and “waves” (because that is what, accordingto the usual interpretation, they really are).

Another meaningful way of retaining the notion of“wave-particle duality” is to understand it as a quantum-classical duality, becuse each classical theory has the cor-responding quantum theory, and vice versa. However,the word “duality” is not the best word for this corre-spondence, because the corresponding quantum and clas-sical theories do not enjoy the same rights. Instead, theclassical theories are merely approximations of the quan-tum ones.

B. Can wave-particle duality be taken seriously?

However, is it possible that the “wave-particle dual-ity” has a literal meaning; that, in some sense, electronsand photons really are both particles and waves? Mostexperts for foundations of QM will probably say – no!Nevertheless, such a definite “no” is also an unprovedmyth. Of course, such a definite “no” is correct if it refersonly to the usual formulation of QM. But who says thatthe usual formulation of QM is the ultimate theory thatwill never be superseded by an even better theory? (Agood scientist will never say that for any theory.) In fact,such a modification of the usual quantum theory alreadyexists. I will refer to it as the Bohmian interpretationof QM [1], but it is also known under the names “deBroglie-Bohm” interpretation and “pilot-wave” interpre-tation. (For recent pedagogic expositions of this interpre-tation, see [2, 3], for a pedagogic comparison with otherformulations of QM, see [4], and for un unbiased review ofadvantages and disadvantages of this interpretation, see[5].) This interpretation consists of two equations. Oneis the standard Schrodinger equation that describes thewave-aspect of the theory, while the other is a classical-like equation that describes a particle trajectory. Theequation for the trajectory is such that the force on theparticle depends on the wave function, so that the motionof the particle differs from that in classical physics, which,in turn, can be used to explain all (otherwise strange)quantum phenomena. In this interpretation, both thewave function and the particle position are fundamen-tal entities. If any known interpretation of QM respectsa kind of wave-particle duality, then it is the Bohmianinterpretation. More on this interpretation (which alsoprovides a counterexample to some other myths of QM)will be presented in subsequent sections.

III. IN QM, THERE IS A TIME-ENERGYUNCERTAINTY RELATION

A. The origin of a time-energy uncertainty relation

For simplicity, consider a particle moving in one dimen-sion. In QM, operators corresponding to the position xand the momentum p satisfy the commutation relation

[x, p] = ih, (2)

where [A,B] ≡ AB−BA. As is well known, this commu-tation relation implies the position-momentum Heisen-berg uncertainty relation

∆x∆p ≥ h

2. (3)

It means that one cannot measure both the particle mo-mentum and the particle position with arbitrary accu-racy. For example, the wave function correponding toa definite momentum is an eigenstate of the momentumoperator

p = −ih ∂∂x. (4)

It is easy to see that such a wave function must be pro-portional to a plane wave eipx/h. On the other hand, thewave function corresponding to an eigenstate of the po-sition operator is essentially a δ-function (see (1)). It isclear that a wave function cannot be both a plane waveand a δ-function, which, in the usual formulation of QM,explains why one cannot measure both the momentumand the position with perfect accuracy.

There is a certain analogy between the couple position-momentum and the couple time-energy. In particular, awave function that describes a particle with a definiteenergy E is proportional to a plane wave e−iEt/h. Anal-ogously, one may imagine that a wave function corre-sponding to a definite time is essentially a δ-function intime. In analogy with (3), this represents an essence ofthe reason for writing the time-energy uncertainty rela-tion

∆t∆E ≥ h

2. (5)

In introductory textbooks on QM, as well as in populartexts on QM, the time-energy uncertainty relation (5)is often presented as a fact enjoying the same rights asthe position-momentum uncertainty relation (3). Nev-ertheless, there is a great difference between these twouncertainty relations. Whereas the position-momentumuncertainty relation (3) is a fact, the time-energy uncer-tainty relation (5) is a myth!

B. The time-energy uncertainty relation is notfundamental

Where does this difference come from? The main dif-ference lies in the fact that energy is not represented by

4

an operator analogous to (4), i.e., energy is not repre-sented by the operator ih∂/∂t. Instead, energy is rep-resented by a much more complicated operator calledHamiltonian, usually having the form

H =p2

2m+ V (x). (6)

Nothing forbids the state ψ(x, t) to be an eigenstate of

H at a definite value of t. This difference has a deeperorigin in the fundamental postulates of QM, according towhich quantum operators are operators on the space offunctions depending on x, not on the space of functionsdepending on t. Thus, space and time have very differentroles in nonrelativistic QM. While x is an operator, t isonly a classical-like parameter. A total probability thatmust be equal to 1 is an integral of the form

∫ ∞

−∞dxψ∗(x, t)ψ(x, t), (7)

not an integral of the form∫ ∞

−∞

∫ ∞

−∞dx dt ψ∗(x, t)ψ(x, t). (8)

In fact, if ψ(x, t) is a solution of the Schrodinger equation,then, when the integral (7) is finite, the integral (8) isnot finite. An analogous statement is also true for one ormore particles moving in 3 dimensions; the probabilitydensity ψ∗ψ is not to be integrated over time.

As the time t is not an operator in QM, a commuta-tion relation analogous to (2) but with the replacementsx→ t, p→ H , does not make sense. This is another rea-son why the time-energy uncertainty relation (5) is notreally valid in QM. Nevertheless, there are attempts toreplace the parameter t with another quantity T , so thatan analog of (2)

[T , H ] = −ih (9)

is valid. However, there is a theorem due to Pauli thatsays that this is impossible [6]. The two main assump-tions of the Pauli theorem are that T and H must be her-mitian operators (because only hermitian operators havereal eigenvalues corresponding to real physical quantities)and that the spectrum of H must be bounded from be-low (which corresponds to the physical requirement thatenergy should not have the possibility to become arbi-trarily negative, because otherwise such a system wouldnot be physically stable). Note that p, unlike H , doesnot need to be bounded from below. For a simple proofof the Pauli theorem, consider the operator

H ′ ≡ e−iǫT /hHeiǫT /h, (10)

where ǫ is a positive parameter with the dimension ofenergy. It is sufficient to consider the case of small ǫ, so,by expanding the exponential functions and using (9),one finds

H ′ ≈ H − ǫ. (11)

Now assume that the spectrum of H is bounded frombelow, i.e., that there exists a ground state |ψ0〉 with

the property H |ψ0〉 = E0|ψ0〉, where E0 is the minimalpossible energy. Consider the state

|ψ〉 = eiǫT /h|ψ0〉. (12)

Assuming that T is hermitian (i.e., that T † = T ) andusing (10) and (11), one finds

〈ψ|H |ψ〉 = 〈ψ0|H ′|ψ0〉 ≈ E0 − ǫ < E0. (13)

This shows that there exists a state |ψ〉 with the en-ergy smaller than E0. This is in contradiction with theasumption that E0 is the minimal energy, which provesthe theorem! There are attempts to modify some of theaxioms of the standard form of quantum theory so thatthe commutation relation (9) can be consistently intro-duced (see, e.g., [7, 8] and references therein), but theviability of such modified axioms of QM is not widelyaccepted among experts.

Although (5) is not a fundamental relation, in mostpractical situations it is still true that the uncertainty∆E and the duration of the measurement process ∆troughly satisfy the inequality (5). However, there existsalso an explicit counterexample that demonstrates that itis possible in principle to measure energy with arbitraryaccuracy during an arbitrarily short time-interval [9].

While different roles of space and time should not besurprising in nonrelativistic QM, one may expect thatspace and time should play a more symmetrical role inrelativistic QM. More on relativistic QM will be said inSec. VII, but here I only note that even in relativisticQM space and time do not play completely symmetricroles, because even there integrals similar to (8) havenot a physical meaning, while those similar to (7) have.Thus, even in relativistic QM, a time-energy uncertaintyrelation does not play a fundamental role.

IV. QM IMPLIES THAT NATURE ISFUNDAMENTALLY RANDOM

A. Fundamental randomness as a myth

QM is a theory that gives predictions on probabilitiesfor different outcomes of measurements. But this is not aprivileged property of QM, classical statistical mechanicsalso does this. Nevertheless, there is an important dif-ference between QM and classical statistical mechanics.The latter is known to be an effective approximative the-ory useful when not all fundamental degrees of freedomare under experimental or theoretical control, while theunderlying more fundamental classical dynamics is com-pletely deterministic. On the other hand, the usual formof QM does not say anything about actual deterministiccauses that lie behind the probabilistic quantum phenom-ena. This fact is often used to claim that QM implies that

5

nature is fundamentally random. Of course, if the usualform of QM is really the ultimate truth, then it is truethat nature is fundamentally random. But who says thatthe usual form of QM really is the ultimate truth? (Aserious scientist will never claim that for any current the-ory.) A priori, one cannot exclude the existence of somehidden variables (not described by the usual form of QM)that provide a deterministic cause for all seemingly ran-dom quantum phenomena. Indeed, from the experiencewith classical pseudorandom phenomena, the existence ofsuch deterministic hidden variables seems a very naturalhypothesis. Nevertheless, QM is not that cheap; in QMthere exist rigorous no-hidden-variable theorems. Thesetheorems are often used to claim that hidden variablescannot exist and, consequently, that nature is fundamen-tally random. However, each theorem has assumptions.The main assumption is that hidden variables must re-produce the statistical predictions of QM. Since thesestatistical predictions are verified experimentally, one isnot allowed to relax this assumption. However, this as-sumption alone is not sufficient to provide a theorem. Inthe actual constructions of these theorems, there are alsosome additional“auxiliary” assumptions, which, however,turn out to be physically crucial! Thus, what these the-orems actually prove, is that hidden variables, if exist,cannot have these additional assumed properties. Sincethere is no independent proof that these additional as-sumed properties are necessary ingredients of nature, theassumptions of these theorems may not be valid. (I shalldiscuss one version of these theorems in more detail inSec. V.) Therefore, the claim that QM implies funda-mental randomness is a myth.

B. From analogy with classical statisticalmechanics to the Bohmian interpretation

Some physicists, including one winner of the Nobelprize [10], very seriously take the possibility that somesort of deterministic hidden variables may underlie theusual form of QM. In fact, the best known and most suc-cessful hidden-variable extension of QM, the Bohmianinterpretation, emerges rather naturally from the anal-ogy with classical statistical mechanics. To see this,consider a classical particle the position of which is notknown with certainty. Instead, one deals with a sta-tistical ensemble in which only the probability densityρ(x, t) is known. The probability must be conserved,i.e.,

∫

d3xρ = 1 for each t. Therefore, the probabilitymust satisfy the local conservation law (known also asthe continuity equation)

∂tρ+ ∇(ρv) = 0, (14)

where v(x, t) is the velocity of the particle at the positionx and the time t. In the Hamilton-Jacobi formulation ofclassical mechanics, the velocity can be calculated as

v(x, t) =∇S(x, t)

m, (15)

where S(x, t) is a solution of the Hamilton-Jacobi equa-tion

(∇S)2

2m+ V (x, t) = −∂tS, (16)

V (x, t) is an arbitrary potential, and m is the mass ofthe particle. The independent real equations (14) and(16) can be written in a more elegant form as a singlecomplex equation. For that purpose, one can introducea complex function [11]

ψ =√ρeiS/h, (17)

where h is an arbitrary constant with the dimension of ac-tion, so that the exponent in (17) is dimensionless. Withthis definition of ψ, Eqs. (14) and (16) are equivalent tothe equation

(−h2∇2

2m+ V −Q

)

ψ = ih∂tψ, (18)

where

Q ≡ − h2

2m

∇2√ρ√ρ. (19)

Indeed, by inserting (17) into (18) and multiplying byψ∗, it is straightforward to check that the real part ofthe resulting equation leads to (16), while the imaginarypart leads to (14) with (15).

The similarity of the classical equation (18) with thequantum Schrodinger equation

(−h2∇2

2m+ V

)

ψ = ih∂tψ (20)

is obvious and remarkable! However, there are also somedifferences. First, in the quantum case, the constant his not arbitrary, but equal to the Planck constant di-vided by 2π. The second difference is the fact that (18)contains the Q-term that is absent in the quantum case(20). Nevertheless, the physical interpretations are thesame; in both cases, |ψ(x, t)|2 is the probability densityof particle positions. On the other hand, we know thatclassical mechanics is fundamentally deterministic. Thisis incoded in the fact that Eq. (18) alone does not providea complete description of classical systems. Instead, oneis also allowed to use Eq. (15), which says that the ve-locity of the particle is determined whenever its positionis also determined. The classical interpretation of thisis that a particle always has a definite position and ve-locity and that the initial position and velocity uniquelydetermine the position and velocity at any time t. Fromthis point of view, nothing seems more natural than toassume that an analogous statement is true also in thequantum case. This assumption represents the core ofthe Bohmian deterministic interpretation of QM. To seethe most obvious consequence of such a classical-like in-terpretation of the Schrodinger equation, note that the

6

Schrodinger equation (20) corresponds to a Hamilton-Jacobi equation in which V in (16) is replaced by V +Q.This is why Q is often referred to as the quantum poten-tial. The quantum potential induces a quantum force.Thus, a quantum particle trajectory satisfies a modifiedNewton equation

md2x

dt2= −∇(V +Q). (21)

Such modified trajectories can be used to explain other-wise strange-looking quantum phenomena (see, e.g., [3]),such as a two-slit experiment.

C. Random or deterministic?

As we have seen above, the analogy between classicalstatistical mechanics and QM can be used to interpretQM in a deterministic manner. However, this analogydoes not prove that such a deterministic interpretationof QM is correct. Indeed, such deterministic quantumtrajectories have never been directly observed. On theother hand, the Bohmian interpretation can explain whythese trajectories are practically unobservable [12], so thelack of experimental evidence does not disprove this in-terpretation. Most experts familiar with the Bohmianinterpretation agree that the observable predictions ofthis interpretation are consistent with those of the stan-dard interpretation, but they often prefer the standardinterpretation because the standard interpretation seemssimpler to them. This is because the standard interpre-tation of QM does not contain Eq. (15). I call this tech-nical simplicity. On the other hand, the advocates of theBohmian interpretation argue that this technical exten-sion of QM makes QM simpler on the conceptual level.Nevertheless, it seems that most contemporary physicistsconsider technical simplicity more important than con-ceptual simplicity, which explains why most physicistsprefer the standard purely probabilistic interpretation ofQM. In fact, by applying a QM-motivated technical cri-terion of simplicity, it can be argued that even classicalstatistical mechanics represented by (18) can be consid-ered complete, in which case even classical mechanics canbe interpreted as a purely probabilistic theory [13]. Butthe fact is that nobody knows with certainty whether thefundamental laws of nature are probabilistic or determin-istic.

V. QM IMPLIES THAT THERE IS NOREALITY BESIDES THE MEASURED REALITY

This is the central myth in QM and many other mythsare based on this one. Therefore, it deserves a particu-larly careful analysis.

A. QM as the ultimate scientific theory?

On one hand, the claim that “there is no reality besidesthe measured reality” may seem to lie at the heart of thescientific method. All scientists agree that the empiricalevidence is the ultimate criterion for acceptance or rejec-tion of any scientific theory, so, from this point of view,such a claim may seem rather natural. On the otherhand, most scientists (apart from quantum physicists)do not find such a radical interpretation of the scientificmethod appealing. In particular, many consider such aninterpretation too antropomorfic (was there any realitybefore humans or living beings existed?), while the his-tory of science surprised us several times by discoveringthat we (the human beings) are not so an important partof the universe as we thought we were. Some quantumphysicists believe that QM is so deep and fundamentalthat it is not just a science that merely applies alreadyprescribed scientific methods, but the science that an-swers the fundamental ontological and epistemologicalquestions on the deepest possible level. But is such a(certainly not modest) belief really founded? What arethe true facts from which such a belief emerged? Let ussee!

B. From a classical variable to a quantumlikerepresentation

Consider a simple real physical classical variable sthat can attain only two different values, say s1 = 1and s2 = −1. By assumption, such a variable cannotchange continuously. Neverheless, a quantity that canstill change continuously is the probability pn(t) that, ata given time t, the variable attains the value sn. Theprobabilities must satisfy

p1(t) + p2(t) = 1. (22)

The average value of s is given by

〈s〉 = s1p1(t) + s2p2(t). (23)

Although s can attain only two values, s1 = 1 and s2 =−1, the average value of s can continuously change withtime and attain an arbitrary value between 1 and −1.

The probabilities pn must be real and non-negative. Asimple formal way to provide this is to write pn = ψ∗

nψn,where ψn are auxiliary quantities that may be negative oreven complex. It is also convenient to view the numbersψn (or ψ∗

n) as components of a vector. This vector canbe represented either as a column

|ψ〉 ≡(

ψ1

ψ2

)

, (24)

or a row

〈ψ| ≡ (ψ∗1 , ψ

∗2). (25)

7

The norm of this vector is

〈ψ|ψ〉 = ψ∗1ψ1 + ψ∗

2ψ2 = p1 + p2. (26)

Thus, the constraint (22) can be viewed as a constrainton the norm of the vector – the norm must be unit. Byintroducing two special unit vectors

|φ1〉 = |↑ 〉 ≡(

10

)

, |φ2〉 = |↓ 〉 ≡(

01

)

, (27)

one finds that the probabilities can be expressed in termsof vector products as

p1 = |〈φ1|ψ〉|2, p2 = |〈φ2|ψ〉|2. (28)

It is also convenient to introduce a diagonal matrix σthat has the values sn at the diagonal and the zeros atall other places:

σ ≡(

s1 00 s2

)

=

(

1 00 −1

)

. (29)

The special vectors (27) have the property

σ|φ1〉 = s1|φ1〉, σ|φ2〉 = s2|φ2〉, (30)

which shows that (i) the vectors (27) are the eigenvectorsof the matrix σ and (ii) the eigenvalues of σ are the al-lowed values s1 and s2. The average value (23) can thenbe formally written as

〈s〉 = 〈ψ(t)|σ|ψ(t)〉. (31)

What has all this to do with QM? First, a discretespectrum of the allowed values is typical of quantum sys-tems; after all, discrete spectra are often referred to as“quantized” spectra, which, indeed, is why QM attainedits name. (Note, however, that it would be misleadingto claim that quantized spectra is the most fundamentalproperty of quantum systems. Some quantum variables,such as the position of a particle, do not have quantizedspectra.) A discrete spectrum contradicts some commonprejudices on classical physical systems because such aspectrum does not allow a continuous change of the vari-able. Nevertheless, a discrete spectrum alone does notyet imply quantum physics. The formal representationof probabilities and average values in terms of complexnumbers, vectors, and matrices as above is, of course,inspired by the formalism widely used in QM; yet, thisrepresentation by itself does not yet imply QM. The for-mal representation in terms of complex numbers, vectors,and matrices can still be interpreted in a classical man-ner.

C. From the quantumlike representation toquantum variables

The really interesting things that deviate significantlyfrom the classical picture emerge when one recalls the fol-lowing formal algebraic properties of vector spaces: Con-sider an arbitrary 2 × 2 unitary matrix U , U †U = 1.

(In particular, U may or may not be time dependent.)Consider a formal transformation

|ψ′〉 = U |ψ〉, 〈ψ′| = 〈ψ|U †,

σ′ = UσU †. (32)

(This transformation refers to all vectors |ψ〉 or 〈ψ|, in-cluding the eigenvectors |φ1〉 and |φ2〉.) In the theory ofvector spaces, such a transformation can be interpretedas a new representation of the same vectors. Indeed, sucha transformation does not change the physical properties,such as the norm of the vector (26), the probabilities (28),and the eigenvalues in (30), calculated in terms of theprimed quantities |ψ′〉, 〈ψ′|, and σ′. This means that theexplicit representations, such as those in (24), (25), (27),and (29), are irrelevant. Instead, the only physically rel-evant properties are abstract, representation-independentquantities, such as scalar products and the spectrum ofeigenvalues. What does it mean physically? One possi-bility is not to take it too seriously, as it is merely an arte-fact of an artificial vector-space representation of certainphysical quantities. However, the history of theoreticalphysics teaches us that formal mathematical symmetriesoften have a deeper physical message. So let us try totake it seriously, to see where it will lead us. Since therepresentation is not relevant, it is natural to ask if thereare other matrices (apart from σ) that do not have theform of (29), but still have the same spectrum of eigen-values as σ? The answer is yes! But then we are in a verystrange, if not paradoxical, position; we have started witha consideration of a single physical variable s and arrivedat a result that seems to suggest the existence of someother, equally physical, variables. As we shall see, thisstrange result lies at the heart of the (also strange) claimthat there is no reality besides the measured reality. Butlet us not jump to the conclusion too early! Instead, letus first study the mathematical properties of these addi-tional physical variables.

Since an arbitrary 2×2 matrix is defined by 4 indepen-dent numbers, each such matrix can be written as a linearcombination of 4 independent matrices. One convenientchoice of 4 independent matrices is

1 ≡(

1 00 1

)

, σ1 ≡(

0 11 0

)

,

σ2 ≡(

0 −ii 0

)

, σ3 ≡(

1 00 −1

)

. (33)

The matrix σ3 is nothing but a renamed matrix σ in(29). The matrices σi, known also as Pauli matrices, arechosen so that they satisfy the familiar symmetricallylooking commutation relations

[σj , σk] = 2iǫjklσl (34)

(where summation over repeated indices is understood).

The matrices σi are all hermitian, σ†i = σi, which im-

plies that their eigenvalues are real. Moreover, all threeσi have the eigenvalues 1 and −1. The most explicit way

8

to see this is to construct the corresponding eigenvectors.The eigenvectors of σ3 are |↑3〉 ≡ |↑ 〉 and |↓3〉 ≡ |↓ 〉 de-fined in (27), with the eigenvalues 1 and −1, respectively.Analogously, it is easy to check that the eigenvectors ofσ1 are

|↑1〉 = 1√2

(

11

)

=|↑ 〉 + |↓ 〉√

2,

|↓1〉 = 1√2

(

1−1

)

=|↑ 〉 − |↓ 〉√

2, (35)

with the eigenvalues 1 and −1, respectively, while theeigenvectors of σ2 are

|↑2〉 = 1√2

(

1i

)

=|↑ 〉 + i|↓ 〉√

2,

|↓2〉 = 1√2

(

1−i

)

=|↑ 〉 − i|↓ 〉√

2, (36)

with the same eigenvalues 1 and −1, respectively.The commutation relations (34) are invariant under

the unitary transformations σi → σ′i = UσiU

†. Thissuggests that the commutation relations themselves aremore physical than the explicit representation given by(33). Indeed, the commutation relations (34) can be rec-ognized as the algebra of the generators of the groupof rotations in 3 spacial dimensions. There is nothingquantum mechanical about that; in classical physics, ma-trices represent operators, that is, abstract objects thatact on vectors by changing (in this case, rotating) them.However, in the usual formulation of classical physics,there is a clear distinction between operators and physi-cal variables – the latter are not represented by matrices.In contrast, in our formulation, the matrices σi have adouble role; mathematically, they are operators (becausethey act on vectors |ψ〉), while physically, they representphysical variables. From symmetry, it is natural to as-sume that all three σi variables are equally physical. Thisassumption is one of the central assumptions of QM thatmakes it different from classical mechanics. For example,the spin operator of spin 1

2 particles in QM is given by

Si =h

2σi, (37)

where the 3 labels i = 1, 2, 3 correspond to 3 space di-rections x, y, z, respectively. Thus, in the case of spin,it is clear that σ3 ≡ σz cannot be more physical thanσ1 ≡ σx or σ2 ≡ σy, despite the fact that σ3 correspondsto the initial physical variable with which we started ourconsiderations. On the other hand, the fact that thenew variables σ1 and σ2 emerged from the initial vari-able σ3 suggests that, in some sense, these 3 variablesare not really completely independent. Indeed, a non-trivial relation among them is incoded in the nontrivialcommutation relations (34). In QM, two variables arereally independent only if their commutator vanishes.(For example, recall that, unlike (34), the position op-erators xi and the momentum operators pi in QM satisfy

[xi, xj ] = [pi, pj ] = 0. In fact, this is the ultimate reasonwhy the most peculiar aspects of QM are usually dis-cussed on the example of spin variables, rather than onposition or momentum variables.)

D. From quantum variables to quantummeasurements

Now consider the state

|ψ〉 =|↑ 〉 + |↓ 〉√

2. (38)

In our initial picture, this state merely represents a situ-ation in which there are 50 : 50 chances that the systemhas the value of s equal to either s = 1 or s = −1. In-deed, if one performs a measurement to find out whatthat value is, one will obtain one and only one of thesetwo values. By doing such a measurement, the observergaines new information about the system. For example,if the value turns out to be s = 1, this gain of informationcan be described by a “collapse”

|ψ〉 → |↑ 〉, (39)

as the state | ↑ 〉 corresponds to a situation in which oneis certain that s = 1. At this level, there is nothingmysterious and nothing intrinsically quantum about thiscollapse. However, in QM, the state (38) contains moreinformation than said above! (Otherwise, there would beno physical difference between the two different states in(35).) From (35), we see that the “uncertain” state (38)corresponds to a situation in which one is absolutely cer-tain that the value of the variable σ1 is equal to 1. On theother hand, if one performs the measurement of s = σ3

and obtains the value as in (39), then the postmeasure-ment state

|↑ 〉 =|↑1〉 + |↓1〉√

2(40)

implies that the value of σ1 is no longer known with cer-tainty. This means that, in some way, the measurementof σ3 destroys the information on σ1. But the crucialquestion is not whether the information on σ1 has beendestroyed, but rather whether the value itself of σ1 hasbeen destroyed. In other words, is it possible that all thetime, irrespective of the performed measurements, σ1 hasthe value 1? The fact is that if this were the case, thenit would contradict the predictions of QM! The simplestway to see this is to observe that, after the first mea-surement with the result (39), one can perform a newmeasurement, in which one measures σ1. From (40), onesees that there are 50% chances that the result of thenew measurement will give the value −1. That is, thereis 0.5 ·0.5 = 0.25 probability that the sequence of the twomeasurements will correspond to the collapses

|↑1〉 → |↑ 〉 → |↓1〉. (41)

9

In (41), the initial value of σ1 is 1, while the final valueof σ1 is −1. Thus, QM predicts that the value of σ1

may change during the process of the two measurements.Since the predictions of QM are in agreement with ex-periments, we are forced to accept this as a fact. Thisdemonstrates that QM is contextual, that is, that themeasured values depend on the context, i.e., on the mea-surement itself. This property by itself is still not in-trinsically quantum, in classical physics the result of ameasurement may also depend on the measurement. In-deed, in classical mechanics there is nothing mysteriousabout it; there, a measurement is a physical process that,as any other physical process, may influence the values ofthe physical variables. But can we talk about the valueof the variable irrespective of measurements? From apurely experimental point of view, we certainly cannot.Here, however, we are talking about theoretical physics.So does the theory allow to talk about that? Classicaltheory certainly does. But what about QM? If all the-oretical knowledge about the system is described by thestate |ψ〉, then quantum theory does not allow that! Thisis the fact. But, can we be sure that we shall never dis-cover some more complete theory than the current formof QM, so that this more complete theory will talk abouttheoretical values of variables irrespective of measure-ments? From the example above, it is not possible todraw such a conclusion. Nevertheless, physicists are try-ing to construct more clever examples from which sucha conclusion could be drawn. Such examples are usuallyreferred to as “no-hidden-variable theorems”. But whatdo these theorems really prove? Let us see!

E. From quantum measurements tono-hidden-variable theorems

To find such an example, consider a system consistingof two independent subsystems, such that each subsys-tem is characterized by a variable that can attain onlytwo values, 1 and −1. The word “independent” (whichwill turn out to be the crucial word) means that the cor-responding operators commute and that the Hamiltoniandoes not contain an interaction term between these twovariables. For example, this can be a system with twofree particles, each having spin 1

2 . In this case, the state|↑ 〉⊗ |↓ 〉 ≡ |↑ 〉|↓ 〉 corresponds to the state in which thefirst particle is in the state |↑ 〉, while the second particleis in the state | ↓ 〉. (The commutativity of the corre-sponding variables is provided by the operators σj ⊗ 1and 1 ⊗ σk that correspond to the variables of the firstand the second subsystem, respectively.) Instead of (38),consider the state

|ψ〉 =|↑ 〉|↓ 〉 + |↓ 〉|↑ 〉√

2. (42)

This state constitutes the basis for the famous Einstein-Podolsky-Rosen-Bell paradox. This state says that if thefirst particle is found in the state | ↑ 〉, then the second

particle will be found in the state |↓ 〉, and vice versa. Inother words, the second particle will always take a direc-tion opposite to that of the first particle. In an oversim-plified version of the paradox, one can wonder how thesecond particle knows about the state of the first par-ticle, given the assumption that there is no interactionbetween the two particles? However, this oversimplifiedversion of the paradox can be easily resolved in classicalterms by observing that the particles do not necessarilyneed to interact, because they could have their (mutu-ally opposite) values all the time even before the mea-surement, while the only role of the measurement was toreveal these values. The case of a single particle discussedthrough Eqs. (38)-(41) suggests that a true paradox canonly be obtained when one assumes that the variablecorresponding to σ1 or σ2 is also a physical variable. In-deed, this is what has been obtained by Bell [14]. Theparadox can be expressed in terms of an inequality thatthe correlation functions among different variables mustobey if the measurement is merely a revealation of thevalues that the noninteracting particles had before themeasurement. (For more detailed pedagogic expositions,see [15, 16].) The predictions of QM turn out to be incontradiction with this Bell inequality. The experimentsviolate the Bell inequality and confirm the predictions ofQM (see [17] for a recent review). This is the fact! How-ever, instead of presenting a detailed derivation of theBell inequality, for pedagogical purposes I shall presenta simpler example that does not involve inequalities, butleads to the same physical implications.

The first no-hidden-variable theorem without inequali-ties has been found by Greenberger, Horne, and Zeilinger[18] (for pedagogic expositions, see [16, 19, 20]), for asystem with 3 particles. However, the simplest such the-orem is the one discovered by Hardy [21] (see also [22])that, like the one of Bell, involves only 2 particles. Al-though pedagogic expositions of the Hardy result alsoexist [16, 20, 23], since it still seems not to be widelyknown in the physics community, here I present a verysimple exposition of the Hardy result (so simple that onecan really wonder why the Hardy result was not discov-ered earlier). Instead of (42), consider a slightly morecomplicated state

|ψ〉 =|↓ 〉|↓ 〉 + |↑ 〉|↓ 〉 + |↓ 〉|↑ 〉√

3. (43)

Using (35), we see that this state can also be written intwo alternative forms as

|ψ〉 =

√2|↓ 〉|↑1 〉 + |↑ 〉|↓ 〉√

3, (44)

|ψ〉 =

√2|↑1〉|↓ 〉 + |↓ 〉|↑ 〉√

3. (45)

From these 3 forms of |ψ〉, we can infer the following:(i) From (43), at least one of the particles is in the state

10

|↓ 〉.(ii) From (44), if the first particle is in the state |↓ 〉, thenthe second particle is in the state |↑1 〉.(iii) From (45), if the second particle is in the state | ↓ 〉,then the first particle is in the state |↑1 〉.Now, by classical reasoning, from (i), (ii), and (iii) oneinfers that(iv) It is impossible that both particles are in the state|↓1 〉.But is (iv) consistent with QM? If it is, then 〈↓1 |〈↓1 |ψ〉must be zero. However, using (44), 〈↓1 | ↑1〉 = 0, andan immediate consequence of (35) 〈↓1 | ↑〉 = −〈↓1 | ↓〉 =

1/√

2, we see that

〈↓1 |〈↓1 |ψ〉 =〈↓1 |↑ 〉〈↓1 |↓ 〉√

3=

−1

2√

3, (46)

which is not zero. Therefore, (iv) is wrong in QM; thereis a finite probability for both particles to be in the state| ↓1〉. This is the fact! But what exactly is wrong withthe reasoning that led to (iv)? The fact is that there areseveral(!) possibilities. Let us briefly discuss them.

F. From no-hidden-variable theorems to physicalinterpretations

One possibility is that classical logic cannot be used inQM. Indeed, this motivated the development of a branchof QM called quantum logic. However, most physicists(as well as mathematicians) consider a deviation fromclassical logic too radical.

Another possibility is that only one matrix, say σ3,corresponds to a genuine physical variable. In this case,the true state of a particle can be | ↑ 〉 or | ↓ 〉, but nota state such as | ↑1〉 or | ↓1〉. Indeed, such a possibilitycorresponds to our starting picture in which there is onlyone physical variable called s that was later artificallyrepresented by the matrix σ ≡ σ3. Such an interpretationmay seem reasonable, at least for some physical variables.However, if σ3 corresponds to the spin in the z-direction,then it does not seem reasonable that the spin in the z-direction is more physical than that in the x-direction orthe y-direction. Picking up one preferred variable breaksthe symmetry, which, at least in some cases, does notseem reasonable.

The third possibility is that one should distinguish be-tween the claims that “the system has a definite value ofa variable” and “the system is measured to have a def-inite value of a variable”. This interpretation of QM iswidely accepted. According to this interpretation, theclaims (i)-(iii) refer only to the results of measurements.These claims assume that σ = σ3 is measured for at leastone of the particles. Consequently, the claim (iv) is validonly if this assumption is fulfilled. In contrast, if σ3 is notmeasured at all, then it is possible to measure both par-ticles to be in the state | ↓1〉. Thus, the paradox that(iv) seems to be both correct and incorrect is merelya manifestation of quantum contextuality. In fact, all

no-hidden-variable theorems can be viewed as manifes-tations of quantum contextuality. However, there are atleast two drastically different versions of this quantum-contextuality interpretation. In the first version, it doesnot make sense even to talk about the values that arenot measured. I refer to this version as the orthodox in-terpretation of QM. (The orthodox interpretation can befurther divided into a hard version in which it is claimedthat such unmeasured values simply do not exist, and asoft version according to which such values perhaps mightexist, but one should not talk about them because onecannot know about the existence of something that isnot measured.) In the second version, the variables havesome values even when they are not measured, but theprocess of measurement is a physical process that mayinfluence these values. The second version assumes thatthe standard formalism of QM is not complete, i.e., thateven a more accurate description of physical systems ispossible than that provided by standard QM. Accordingto this version, “no-hidden-variable” theorems (such asthe one of Bell or Hardy) do not really prove that hiddenvariables cannot exist, because these theorems assumethat there are no interactions between particles, whilethis assumption may be violated at the level of hiddenvariables.

Most pragmatic physicists seem to (often tacitly) ac-cept the soft-orthodox interpretation. From a pragmaticpoint of view, such an attitude seems rather reasonable.However, physicists who want to understand QM at thedeepest possible level can hardly be satisfied with the softversion of the orthodox interpretation. They are forcedeither to adopt the hard-orthodox interpretation or tothink about the alternatives (like hidden variables, pre-ferred variables, or quantum logic). Among these physi-cists that cope with the foundations of QM at the deepestlevel, the hard-orthodox point of view seems to domi-nate. (If it did not dominate, then I would not call it“orthodox”). However, even the advocates of the hard-orthodox interpretation do not really agree what exactlythis interpretation means. Instead, there is a numberof subvariants of the hard-orthodox interpretation thatdiffer in the fundamental ontology of nature. Some ofthem are rather antropomorphic, by attributing a fun-damental role to the observers. However, most of themattempt to avoid antropomorphic ontology, for exampleby proposing that the concept of information on reality ismore fundamental than the concept of reality itself [24],or that reality is relative or “relational” [25, 26], or thatcorrelations among variables exist, while the variablesthemselves do not [27]. Needless to say, all such versionsof the hard-orthodox interpretation necessarily involvedeep (and dubious) philosophical assumptions and pos-tulates. To avoid philosophy, an alternative is to adopta softer version of the orthodox interpretation (see, e.g.,[28]). The weakness of the soft versions is the fact thatthey do not even try to answer fundamental questionsone may ask, but their advocates often argue that thesequestions are not physical, but rather metaphysical or

11

philosophical.Let us also discuss in more detail the possibility that

one variable is more physical than the others, that onlythis preferred variable corresponds to the genuine phys-ical reality. Of course, it does not seem reasonable thatspin in the z-direction is more physical than that in thex- or the y-direction. However, it is not so unreasonablethat, for example, the particle position is a more funda-mental variable than the particle momentum or energy.(After all, most physicists will agree that this is so inclassical mechanics, despite the fact that the Hamilto-nian formulation of classical mechanics treats positionand momentum on an equal footing.) Indeed, in prac-tice, all quantum measurements eventually reduce to anobservation of the position of something (such as the nee-dle of the measuring apparatus). In particular, the spinof a particle is measured by a Stern-Gerlach apparatus,in which the magnetic field causes particles with one ori-entation of the spin to change their direction of motionto one side, and those with the opposite direction to theother. Thus, one does not really observe the spin itself,but rather the position of the particle. In general, as-sume that one wants to measure the value of the variabledescribed by the operator A. It is convenient to intro-duce an orthonormal basis |ψa〉 such that each |ψa〉 isan eigenvector of the operator A with the eigenvalue a.The quantum state can be expanded in this basis as

|ψ〉 =∑

a

ca|ψa〉, (47)

where (assuming that the spectrum of A is not degen-erate) |ca|2 is the probability that the variable will bemeasured to have the value a. To perform a measure-ment, one must introduce the degrees of freedom of themeasuring apparatus, which, before the measurement, isdescribed by some state |φ〉. In an ideal measurement,the interaction between the measured degrees of freedomand the degrees of freedom of the measuring apparatusmust be such that the total quantum state exhibits en-tanglement between these two degrees of freedom, so thatthe total state takes the form

|Ψ〉 =∑

a

ca|ψa〉|φa〉, (48)

where |φa〉 are orthonormal states of the measuring appa-ratus. Thus, whenever the measuring apparatus is foundin the state |φa〉, one can be certain (at least theoreti-cally) that the state of the measured degree of freedomis given by |ψa〉. Moreover, from (48) it is clear thatthe probability for this to happen is equal to |ca|2, thesame probability as that without introducing the measur-ing apparatus. Although the description of the quantummeasurement as described above is usually not discussedin practical textbooks on QM, it is actually a part of thestandard form of quantum theory and does not dependon the interpretation. (For modern practical introduc-tory lectures on QM in which the theory of measure-ment is included, see, e.g., [29].) What this theory of

quantum measurement suggests is that, in order to re-produce the statistical predictions of standard QM, it isnot really necessary that all hermitian operators called“observables” correspond to genuine physical variables.Instead, it is sufficient that only one or a few preferredvariables that are really measured in practice correspondto genuine physical variables, while the rest of the “ob-servables” are merely hermitian operators that do notcorrespond to true physical reality [30]. This is actuallythe reason why the Bohmian interpretation discussed inthe preceding section, in which the preferred variables arethe particle positions, is able to reproduce the quantumpredictions on all quantum observables, such as momen-tum, energy, spin, etc. Thus, the Bohmian interpretationcombines two possibilities discussed above: one is the ex-istence of the preferred variable (the particle position)and the other is the hidden variable (the particle posi-tion existing even when it is not measured).

To conclude this section, QM does not prove that thereis no reality besides the measured reality. Instead, thereare several alternatives to it. In particular, such real-ity may exist, but then it must be contextual (i.e., mustdepend on the measurement itself.) The simplest (al-though not necessary) way to introduce such reality isto postulate it only for one or a few preferred quantumobservables.

VI. QM IS LOCAL/NONLOCAL

A. Formal locality of QM

Classical mechanics is local. This means that a phys-ical quantity at some position x and time t may be in-fluenced by another physical quantity only if this otherphysical quantity is attached to the same x and t.For example, two spacially separated local objects can-not communicate directly, but only via a third physi-cal object that can move from one object to the other.In the case of n particles, the requirement of localitycan be written as a requirement that the HamiltonianH(x1, . . . ,xn,p1, . . . ,pn) should have the form

H =n∑

l=1

Hl(xl,pl). (49)

In particular, a nontrivial 2-particle potential of the formV (x1 −x2) is forbidden by the principle of locality. Notethat such a potential is not forbidden in Newtonian clas-sical mechanics. However, known fundamental interac-tions are relativistic interactions that do not allow suchinstantaneous communications. At best, such a nonlocalpotential can be used as an approximation valid whenthe particles are sufficiently close to each other and theirvelocities are sufficiently small.

The quantum Hamiltonian is obtained from the cor-responding classical Hamiltonian by a replacement ofclassical positions and momenta by the corresponding

12

quantum operators. Thus, the quantum Hamiltoniantakes the same local form as the classical one. Since theSchrodinger equation

H |ψ(t)〉 = ih∂t|ψ(t)〉 (50)

is based on this local Hamiltonian, any change of thewave function induced by the Schrodinger equation (50)is local. This is the fact. For this reason, it is oftenclaimed that QM is local to the same extent as classicalmechanics is.

B. (Non)locality and hidden variables

The principle of locality is often used as the crucial ar-gument against hidden variables in QM. For example,consider two particles entangled such that their wavefunction (with the spacial and temporal dependence ofwave functions suppressed) takes the form (43). Such aform of the wave function can be kept even when theparticles become spacially separated. As we have seen,the fact that (iv) is inconsistent with QM can be inter-preted as QM contextuality. However, we have seen thatthere are two versions of QM contextuality – the ortho-dox one and the hidden-variable one. The principle oflocality excludes the hidden-variable version of QM con-textuality, because this version requires interactions be-tween the two particles, which are impossible when theparticles are (sufficiently) spacially separated. However,it is important to emphasize that the principle of localityis an assumption. We know that the Schrodinger equa-tion satisfies this principle, but we do not know if thisprinciple must be valid for any physical theory. In par-ticular, subquantum hidden variables might not satisfythis principle. Physicists often object that nonlocal in-teractions contradict the theory of relativity. However,there are several responses to such objections. First, thetheory of relativity is just as any other theory – nobodycan be certain that this theory is absolutely correct atall (including the unexplored ones) levels. Second, non-locality by itself does not necessarily contradict relativ-ity. For example, a local relativistic-covariant field the-ory (see Sec. VIII) can be defined by an action of theform

∫

d4xL(x), where L(x) is the local Lagrangian den-sity transforming (under arbitrary coordinate transfor-mations) as a scalar density. A nonlocal action may havea form

∫

d4x∫

d4x′L(x, x′). If L(x, x′) transforms as abi-scalar density, then such a nonlocal action is relativis-tically covariant. Third, the nonlocality needed to ex-plain quantum contextuality requires instantaneous com-munication, which is often claimed to be excluded by thetheory of relativity, as the velocity of light is the maxi-mal possible velocity allowed by the theory of relativity.However, this is actually a myth in the theory of rela-tivity; this theory by itself does not exclude faster-than-light communication. It excludes it only if some addi-tional assumptions on the nature of matter are used. The

best known counterexample are tachyons [31] – hypo-thetical particles with negative mass squared that movefaster than light and fully respect the theory of relativity.Some physicists argue that faster-than-light communica-tion contradicts the principle of causality, but this is alsonothing but a myth [32, 33]. (As shown in [33], this mythcan be traced back to one of the most fundamental mythsin physics according to which time fundamentally differsfrom space by having a property of “lapsing”.) Finally,some physicists find absurd or difficult even to conceivephysical laws in which information between distant ob-jects is transferred instantaneously. It is ironic that theyprobably had not such mental problems many years agowhen they did not know about the theory of relativitybut did know about the Newton instantaneous law ofgravitation or the Coulomb instantaneous law of electro-statics. To conclude this paragraph, hidden variables, ifexist, must violate the principle of locality, which may ormay not violate the theory of relativity.

To illustrate nonlocality of hidden variables, I considerthe example of the Bohmian interpretation. For a many-particle wave function Ψ(x1, . . . ,xn, t) that describes nparticles with the mass m, it is straightforward to showthat the generalization of (19) is

Q(x1, . . . ,xn, t) = − h2

2m

n∑

l=1

∇2l

√

ρ(x1, . . . ,xn, t)

√

ρ(x1, . . . ,xn, t). (51)

When the wave function exhibits entanglement, i.e.,when Ψ(x1, . . . ,xn, t) is not a local product of the formψ1(x1, t) · · ·ψn(xn, t), then Q(x1, . . . ,xn, t) is not of theform

∑nl=1Ql(xl, t) (compare with (49)). In the Bohmian

interpretation, this means that Q is the quantum poten-tial which (in the case of entanglement) describes a non-local interaction. For attempts to formulate the nonlocalBohmian interaction in a relativistic covariant way, see,e.g., [34, 35, 36, 37, 38, 39].

C. (Non)locality without hidden variables?

Concerning the issue of locality, the most difficult ques-tion is whether QM itself, without hidden variables, is lo-cal or not. The fact is that there is no consensus amongexperts on that issue. It is known that quantum effects,such as the Einstein-Podolsky-Rosen-Bell effect or theHardy effect, cannot be used to transmit information.This is because the choice of the state to which the sys-tem will collapse is random (as we have seen, this ran-domness may be either fundamental or effective), so onecannot choose to transmit the message one wants. Inthis sense, QM is local. On the other hand, the cor-relation among different subsystems is nonlocal, in thesense that one subsystem is correlated with another sub-system, such that this correlation cannot be explained ina local manner in terms of preexisting properties before

13

the measurement. Thus, there are good reasons for theclaim that QM is not local.

Owing to the nonlocal correlations discussed above,some physicists claim that it is a fact that QM is notlocal. Nevertheless, many experts do not agree with thisclaim, so it cannot be regarded as a fact. Of course, atthe conceptual level, it is very difficult to conceive hownonlocal correlations can be explained without nonlocal-ity. Nevertheless, hard-orthodox quantum physicists aretrying to do that (see, e.g., [24, 25, 26, 27]). In order tosave the locality principle, they, in one way or another,deny the existence of objective reality. Without objec-tive reality, there is nothing to be objectively nonlocal.What remains is the wave function that satisfies a localSchrodinger equation and does not represent reality, butonly the information on reality, while reality itself doesnot exist in an objective sense. Many physicists (includ-ing myself) have problems with thinking about informa-tion on reality without objective reality itself, but it doesnot prove that such thinking is incorrect.

To conclude, the fact is that, so far, there has beenno final proof with which most experts would agree thatQM is either local or nonlocal. There is only agreementthat if hidden variables (that is, objective physical prop-erties existing even when they are not measured) exist,then they must be nonlocal. Some experts consider thisa proof that they do not exist, whereas other expertsconsider this a proof that QM is nonlocal. They con-sider these as proofs because they are reluctant to giveup either of the principle of locality or of the existence ofobjective reality. Nevertheless, more open-minded (somewill say – too open-minded) people admit that neither ofthese two “crazy” possibilities (nonlocality and absenceof objective reality) should be a priori excluded.

VII. THERE IS A WELL-DEFINEDRELATIVISTIC QM

A. Klein-Gordon equation and the problem ofprobabilistic interpretation

The free Schrodinger equation

−h2∇2

2mψ(x, t) = ih∂tψ(x, t) (52)

is not consistent with the theory of relativity. In par-ticular, it treats space and time in completely differentways, which contradicts the principle of relativistic co-variance. Eq. (52) corresponds only to a nonrelativisticapproximation of QM. What is the corresponding rela-tivistic equation from which (52) can be derived as anapproximation? Clearly, the relativistic equation musttreat space and time on an equal footing. For that pur-pose, it is convenient to choose units in which the velocityof light is c = 1. To further simplify equations, it is alsoconvenient to further restrict units so that h = 1. Intro-ducing coordinates xµ, µ = 0, 1, 2, 3, where x0 = t, while

x1, x2, x3 are space coordinates, the simplest relativisticgeneralization of (52) is the Klein-Gordon equation

(∂µ∂µ +m2)ψ(x) = 0, (53)

where x = xµ, summation over repeated indices is un-derstood, ∂µ∂µ = ηµν∂µ∂ν , and ηµν is the diagonal met-ric tensor with η00 = 1, η11 = η22 = η33 = −1. However,the existence of this relativistic wave equation does notimply that relativistic QM exists. This is because thereare interpretational problems with this equation. In non-relativistic QM, the quantity ψ∗ψ is the probability den-sity, having the property

d

dt

∫

d3xψ∗ψ = 0, (54)

which can be easily derived from the Schrodinger equa-tion (52). This property is crucial for the consistencyof the probabilistic interpretation, because the integral∫

d3xψ∗ψ is the sum of all probabilities for the particleto be at all possible places, which must be equal to 1 foreach time t. If ψ is normalized so that this integral isequal to 1 at t = 0, then (54) provides that it is equalto 1 at each t. However, when ψ satisfies (53) insteadof (52), then the consistency requirement (54) is not ful-filled. Consequently, in relativistic QM based on (53),ψ∗ψ cannot be interpreted as the probability density.

In order to solve this problem, one can introduce theKlein-Gordon current

jµ = iψ∗ ↔∂µψ, (55)

where a↔∂µ b ≡ a(∂µb)− (∂µa)b. Using (53), one can show

that this current satisfies the local conservation law

∂µjµ = 0, (56)

which implies that

d

dt

∫

d3x j0 = 0. (57)

Eq. (57) suggests that, in the relativistc case, it is j0 thatshould be interpreted as the probability density. Moregenerally, if ψ1(x) and ψ2(x) are two solutions of (53),then the scalar product defined as

(ψ1, ψ2) = i

∫

d3xψ∗1(x)

↔∂0ψ2(x) (58)

does not depend on time. The scalar product (58) doesnot look relativistic covariant, but there is a way to writeit in a relativistic covariant form. The constant-timespacelike hypersurface with the infinitesimal volume d3xcan be generalized to an arbitrarily curved spacelike hy-persurface Σ with the infinitesimal volume dSµ orientedin a timelike direction normal to Σ. Eq. (58) then gener-alizes to

(ψ1, ψ2) = i

∫

Σ

dSµ ψ∗1(x)

↔∂µψ2(x), (59)

14

which, owing to the 4-dimensional Gauss law, does notdepend on Σ when ψ1(x) and ψ2(x) satisfy (53). How-ever, there is a problem again. The general solution of(53) can be written as

ψ(x) = ψ+(x) + ψ−(x), (60)

where

ψ+(x) =∑

k

cke−i(ωkt−kx), (61)

ψ−(x) =∑

k

dkei(ωkt−kx).

Here ck and dk are arbitrary complex coefficients, and

ωk ≡√

k2 +m2 (62)

is the frequency. For obvious reasons, ψ+ is called apositive-frequency solution, while ψ− is called a negative-frequency solution. (The positive- and negative-frequencysolutions are often referred to as positive- and negative-energy solutions, respectively. However, such a termi-nology is misleading because in field theory, which willbe discussed in the next section, energy cannot be nega-tive, so it is better to speak of positive and negative fre-quency.) The nonrelativistic Schrodinger equation con-tains only the positive-frequency solutions, which canbe traced back to the fact that the Schrodinger equa-tion contains a first time derivative, instead of a secondtime derivative that appears in the Klein-Gordon equa-tion (53). For a positive-frequency solution the quan-tity

∫

d3x j0 is positive, whereas for a negative-frequencysolution this quantity is negative. Since the sum of allprobabilities must be positive, the negative-frequency so-lutions represent a problem for the probabilistic interpre-tation. One may propose that only positive-frequencysolutions are physical, but even this does not solve theproblem. Although the integral

∫

d3x j0 is strictly pos-itive in that case, the local density j0(x) may still benegative at some regions of spacetime, provided that thesuperposition ψ+ in (61) contains terms with two or moredifferent positive frequencies. Thus, even with strictlypositive-frequency solutions, the quantity j0 cannot beinterpreted as a probability density.

B. Some attempts to solve the problem

Physicists sometimes claim that there are no interpre-tational problems with the Klein-Gordon equation be-cause the coefficients ck and dk in (61) (which are theFourier transforms of ψ+ and ψ−, respectively) are timeindependent, so the quantities c∗

kck and d∗

kdk can be con-

sistently interpreted as probability densities in the mo-mentum space. (More precisely, if ck and dk are indepen-dent, then these two probability densities refer to parti-cles and antiparticles, respectively.) Indeed, in practicalapplications of relativistic QM, one is often interested

only in scattering processes, in which the probabilities ofdifferent momenta contain all the information that canbe compared with actual experiments. From a practi-cal point of view, this is usually enough. Nevertheless,in principle, it is possible to envisage an experiment inwhich one measures the probabilities in the position (i.e.,configuration) space, rather than that in the momentumspace. A complete theory should have predictions on allquantities that can be measured in principle. Besides,if the standard interpretation of the nonrelativistic wavefunction in terms of the probability density in the po-sition space is correct (which, indeed, is experimentallyconfirmed), then this interpretation must be derivablefrom a more accurate theory – relativistic QM. Thus,the existence of the probabilistic interpretation in themomentum space does not really solve the problem.

It is often claimed that the problem of relativistic prob-abilistic interpretation in the position space is solved bythe Dirac equation. As we have seen, the problems withthe Klein-Gordon equation can be traced back to the factthat it contains a second time derivative, instead of a firstone. The relativistic-covariant wave equation that con-tains only first derivatives with respect to time and spaceis the Dirac equation

(iγµ∂µ −m)ψ(x) = 0. (63)

Here γµ are the 4 × 4 Dirac matrices that satisfy theanticommutation relations

γµ, γν = 2ηµν , (64)

where A,B ≡ AB + BA. The Dirac matrices are re-lated to the Pauli matrices σi discussed in Sec. V, whichsatisfy σi, σj = 2δij . (For more details, see, e.g., [40].)It turns out that ψ in (63) is a 4-component wave func-tion called spinor that describes particles with spin 1

2 .The conserved current associated with (63) is

jµ = ψγµψ, (65)

where ψ ≡ ψ†γ0. In particular, (64) implies γ0γ0 = 1, so(65) implies

j0 = ψ†ψ, (66)

which cannot be negative. Thus, the Dirac equation doesnot have problems with the probabilistic interpretation.However, this still does not mean that the problems ofrelativistic QM are solved. This is because the Diracequation describes only particles with spin 1

2 . Particleswith different spins also exist in nature. In particular,the Klein-Gordon equation describes particles with spin0, while the wave equation for spin 1 particles are essen-tially the Maxwell equations, which are second-order dif-ferential equations for the electromagnetic potential Aµ

and lead to the same interpretational problems as theKlein-Gordon equation.

There are various proposals for a more direct solu-tion to the problem of probabilistic interpretation of the

15

Klein-Gordon equation (see, e.g., [37, 41, 42, 43, 44]).However, all these proposed solutions have certain disad-vantages and none of these proposals is widely acceptedas the correct solution. Therefore, without this problembeing definitely solved, it cannot be said that there existsa well-defined relativistic QM.

VIII. QUANTUM FIELD THEORY SOLVESTHE PROBLEMS OF RELATIVISTIC QM

It is often claimed that the interpretational problemswith relativistic QM discussed in the preceding sectionare solved by a more advanced theory – quantum fieldtheory (QFT). To see how QFT solves these problemsand whether this solution is really satisfactory, let mebriefly review what QFT is and why it was introduced.

A. Second quantization of particles

A theoretical concept closely related to QFT is themethod of second quantization. It was introduced to for-mulate in a more elegant way the fact that many-particlewave functions should be either completely symmetricor completely antisymmetric under exchange of any twoparticles, which comprises the principle that identicalparticles cannot be distinguished. Let

ψ(x, t) =∑

k

akfk(x, t) (67)

be the wave function expanded in terms of some com-plete orthonormal set of solutions fk(x, t). (For free par-ticles, fk(x, t) are usually taken to be the plane wavesfk(x, t) ∝ e−i(ωt−kx).) Unlike the particle position x,the wave function ψ does not correspond to an operator.Instead, it is just an ordinary number that determinesthe probability density ψ∗ψ. This is so in the ordinary“first” quantization of particles. The method of secondquantization promotes the wave function ψ to an opera-

tor ψ. (To avoid confusion, from now on, the operatorsare always denoted by a hat above it.) Thus, instead of(67), we have the operator

ψ(x, t) =∑

k

akfk(x, t), (68)

where the coefficients ak are also promoted to the oper-ators ak. Similarly, instead of the complex conjugatedwave function ψ∗, we have the hermitian conjugated op-erator

ψ†(x, t) =∑

k

a†kf∗k (x, t). (69)

The orthonormal solutions fk(x, t) are still ordinary func-

tions as before, so that the operator ψ satisfies the sameequation of motion (e.g., the Schrodinger equation in the

nonrelativistic case) as ψ. In the case of bosons, the op-

erators ak, a†k are postulated to satisfy the commutation

relations

[ak, a†k′ ] = δkk′ ,

[ak, ak′ ] = [a†k, a†k′ ] = 0. (70)

These commutation relations are postulated because, asis well-known from the case of first-quantized harmonicoscillator (discussed also in more detail in the next sec-tion), such commutation relations lead to a representa-

tion in which a†k and ak are raising and lowering oper-ators, respectively. Thus, an n-particle state with thewave function f(k1, . . . , kn) in the k-space can be ab-stractly represented as

|nf 〉 =∑

k1,...,kn

f(k1, . . . , kn) a†k1· · · a†kn

|0〉, (71)

where |0〉 is the ground state of second quantization, i.e.,the vacuum state containing no particles. Introducingthe operator

N =∑

k

a†kak, (72)

and using (70), one can show that

N |nf 〉 = n|nf〉. (73)

Since n is the number of particles in the state (71),

Eq. (73) shows that N is the operator of the numberof particles. The n-particle wave function in the config-uration space can then be written as

ψ(x1, . . . ,xn, t) = 〈0|ψ(x1, t) · · · ψ(xn, t)|nf 〉. (74)

From (70) and (68) we see that ψ(x, t)ψ(x′, t) =

ψ(x′, t)ψ(x, t), which implies that the ordering of the ψ-operators in (74) is irrelevant. This means that (74) auto-matically represents a bosonic wave function completelysymmetric under any two exchanges of the argumentsxa, a = 1, . . . , n. For the fermionic case, one replacesthe commutation relations (70) with similar anticommu-tation relations

ak, a†k′ = δkk′ ,

ak, ak′ = a†k, a†k′ = 0, (75)

which, in a similar way, leads to completely antisymmet-ric wave functions.

B. Quantum fields

The method of second quantization outlined above isnothing but a convenient mathematical trick. It does notbring any new physical information. However, the math-ematical formalism used in this trick can be reinterpreted

16

in the following way: The fundamental quantum objectis neither the particle with the position-operator x northe wave function ψ, but a new hermitian operator

φ(x, t) = ψ(x, t) + ψ†(x, t). (76)

This hermitian operator is called field and the result-ing theory is called quantum field theory (QFT). It is aquantum-operator version of a classical field φ(x, t). (Aprototype of classical fields is the electromagnetic fieldsatisfying Maxwell equations. Here, for pedagogical pur-poses, we do not study the electromagnetic field, but onlythe simplest scalar field φ.) Using (68), (69), and (70),one obtains

[φ(x, t), φ(x′, t)] =∑

k

fk(x, t)f∗k (x′, t)

−∑

k

f∗k (x, t)fk(x′, t). (77)

Thus, by using the completeness relations

∑

k

fk(x, t)f∗k (x′, t) =

∑

k

f∗k (x, t)fk(x′, t) = δ3(x − x′),

(78)one finally obtains

[φ(x, t), φ(x′, t)] = 0. (79)

Thus, from (68), (69), (70), and (76) one finds that (74)can also be written as

ψ(x1, . . . ,xn, t) = 〈0|φ(x1, t) · · · φ(xn, t)|nf 〉, (80)

which, owing to (79), provides the complete symmetry ofψ.

The field equations of motion are derived from theirown actions. For example, the Klein-Gordon equation(53) for φ(x) (instead of ψ(x)) can be obtained from theclassical action

A =

∫

d4xL, (81)

where

L(φ, ∂αφ) =1

2[(∂µφ)(∂µφ) −m2φ2] (82)

is the Lagrangian density. The canonical momentum as-sociated with this action is a fieldlike quantity

π(x) =∂L

∂(∂0φ(x))= ∂0φ(x). (83)

The associated Hamiltonian density is

H = π∂0φ− L =1

2[π2 + (∇φ)2 +m2φ2]. (84)

This shows that the field energy

H [π, φ] =

∫

d3xH(π(x), φ(x),∇φ(x)) (85)

(where the time-dependence is suppressed) cannot benegative. This is why, in relativistic QM, it is betterto speak of negative frequencies than of negative ener-gies. (In (85), the notation H [π, φ] denotes that H is nota function of π(x) and φ(x) at some particular values ofx, but a functional, i.e., an object that depends on thewhole functions π and φ at all values of x.) By analogywith the particle commutation relations [xl, pm] = iδlm,[xi, xj ] = [pi, pj] = 0, the fundamental field-operatorcommutation relations are postulated to be

[φ(x), π(x′)] = iδ3(x − x′),

[φ(x), φ(x′)] = [π(x), π(x′)] = 0. (86)

Here it is understood that all fields are evaluated at thesame time t, so the t dependence is not written explicitly.Thus, now (79) is one of the fundamental (not derived)

commutation relations. Since φ(x) is an operator in theHeisenberg picture that satisfies the Klein-Gordon equa-tion, the expansion (76) with (68) and (69) can be used.One of the most important things gained from quantiza-tion of fields is the fact that now the commutation rela-tions (70) do not need to be postulated. Instead, theycan be derived from the fundamental field-operator com-mutation relations (86). (The fermionic field-operatorssatisfy similar fundamental relations with commutatorsreplaced by anticommutators, from which (75) can be de-rived.) The existence of the Hamiltonian (85) allows usto introduce the functional Schrodinger equation

H [π, φ]Ψ[φ; t) = i∂tΨ[φ; t), (87)

where Ψ[φ; t) is a functional of φ(x) and a function of t,while

π(x) = −i δ

δφ(x)(88)

is the field analog of the particle-momentum operatorpj = −i∂/∂xj. (For a more careful definition of thefunctional derivative δ/δφ(x) see, e.g., [45].) Unlike theKlein-Gordon equation, the functional Schrodinger equa-tion (87) is a first-order differential equation in the timederivative. Consequently, the quantity

ρ[φ; t) = Ψ∗[φ; t)Ψ[φ; t) (89)

can be consistently interpreted as a conserved probabilitydensity. It represents the probability that the field hasthe configuration φ(x) at the time t.

C. Does QFT solve the problems of relativisticQM?

After this brief overview of QFT, we are finally readyto cope with the validity of the title of this section. HowQFT helps in solving the interpretational problems ofrelativistic QM? According to QFT, the fundamental ob-jects in nature are not particles, but fields. Consequently,

17

the fundamental wave function(al) that needs to havea well-defined probabilistic interpretation is not ψ(x, t),but Ψ[φ; t). Thus, the fact that, in the case of Klein-Gordon equation, ψ(x, t) cannot be interpreted proba-bilistically, is no longer a problem from this more fun-damental point of view. However, does it really solvethe problem? If QFT is really a more fundamental the-ory than the first-quantized quantum theory of particles,then it should be able to reproduce all good results of thisless fundamental theory. In particular, from the funda-mental axioms of QFT (such as the axiom that (89) rep-resents the probability in the space of fields), one shouldbe able to deduce that, at least in the relativistic limit,ψ∗ψ represents the probability in the space of particlepositions. However, one cannot deduce it solely from theaxioms of QFT. One possibility is to completely ignore,or even deny [46], the validity of the probabilistic in-terpretation of ψ, which indeed is in the spirit of QFTviewed as a fundamental theory, but then the problemis to reconcile it with the fact that such a probabilis-tic interpretation of ψ is in agreement with experiments.Another possibility is to supplement the axioms of QFTwith an additional axiom that says that ψ in the non-relativistic limit determines the probabilities of particlepositions, but then such a set of axioms is not coherent,as it does not specify the meaning of ψ in the relativisticcase. Thus, instead of saying that QFT solves the prob-lems of relativistic QM, it is more honest to say that itmerely sweeps them under the carpet.

IX. QUANTUM FIELD THEORY IS A THEORYOF PARTICLES

What is the world made of? An often answer is thatit is made of elementary particles, such as electrons, pho-tons, quarks, gluons, etc. On the other hand, all mod-ern theoretical research in elementary-particle physics isbased on quantum field theory (QFT) [45, 47, 48]. So,is the world made of particles or fields? An often an-swer given by elementary-particle physicists is that QFTis actually a theory of particles, or more precisely, thatparticles are actually more fundamental physical objects,while QFT is more like a mathematical tool that de-scribes – the particles. Indeed, the fact that the mo-tivation for introducing QFT partially emerged from themethod of second quantization (see Sec. VIII) supportsthis interpretation according to which QFT is nothingbut a theory of particles. But is that really so? Is itreally a fundamental property of QFT that it describesparticles? Let us see!

A. A first-quantized analog of particles in QFT

From the conceptual point of view, fields and particlesare very different objects. This is particularly clear forclassical fields and particles, where all concepts are clear.

So, if there exists a relation between quantum fields andparticles that does not have an analog in the classicaltheory of fields and particles, then such a relation mustbe highly nontrivial. Indeed, this nontrivial relation isrelated to the nontrivial commutation relations (70) (or(75) for fermionic fields). The classical fields commute,which implies that the classical coefficients ak, a∗k do notsatisfy (70). Without these commutation relations, wecould not introduce n-particle states (71). However, arethe commutation relations (70) sufficient for having awell-defined notion of particles? To answer this ques-tion, it is instructive to study the analogy with the first-quantized theory of particles.

Consider a quantum particle moving in one dimension,having a Hamiltonian

H =p

2m+ V (x). (90)

We introduce the operators

a = 1√2

(√mωx+ i

p√mω

)

,

a† = 1√2

(√mωx− i

p√mω

)

, (91)

where ω is some constant of the dimension of energy (orfrequency, which, since h = 1, has the same dimension asenergy). Using the commutation relation [x, p] = i, weobtain

[a, a†] = 1. (92)

This, together with the trivial commutation relations[a, a] = [a†, a†] = 0, shows that a† and a are the rais-ing and lowering operator, respectively. As we speak ofone particle, the number operator N = a†a now cannotbe called the number of particles. Instead, we use a moregeneral terminology (applicable to (72) as well) accord-

ing to which N is the number of “quanta”. But quantaof what? It is easy to show that (90) can be written as

H = ω

(

N +1

2

)

+

[

V (x) − mω2x2

2

]

. (93)

In the special case in which V (x) = mω2x2/2, which cor-responds to the harmonic oscillator, the square bracketin (93) vanishes, so the Hamiltonian can be expressed in

terms of the N -operator only. In this case, the (properlynormalized) state

|n〉 =(a†)n

√n!

|0〉 (94)

has the energy ω(n+1/2), so the energy can be viewed asa sum of the ground-state energy ω/2 and the energy of nquanta with energy ω. This is why the number operatorN plays an important physical role. However, the mainpoint that can be inferred from (93) is the fact that, for a

18

general potential V (x), the number operator N does notplay any particular physical role. Although the spectrumof quantum states can often be labeled by a discrete la-bel n′ = 0, 1, 2, . . ., this label, in general, has nothing todo with the operator N (i.e., the eigenstates (94) of Nare not the eigenstates of the Hamiltonian (93)). More-over, in general, the spectrum of energies E(n′) may havea more complicated dependence on n′, so that, unlikethe harmonic oscillator, the spectrum of energies is notequidistant. Thus, in general, the state of a system can-not be naturally specified by a number of “quanta” n.

If one insists on representing the system in terms ofthe states (94), then one can treat the square bracketin (93) as a perturbation VI(x). (Here “I” stands for“interaction”.) From (91) one finds

x =a+ a†√

2mω, (95)

so VI(x) = VI(a, a†). Consequently, various terms in the

perturbation expansion can be represented in terms ofcreation and destruction of quanta, owing to the occur-rence of a† and a, respectively. However, treating thesquare bracket in (93) as a perturbation is completelyarbitrary. Such a treatment is nothing but a mathe-matical convenience and does not make the states (94)more physical. This is particularly clear for the casesin which the original system with the Hamiltonian (90)can be solved analytically, without a perturbation expan-sion. The creation and destruction of quanta appearingin the perturbation expansion does not correspond to ac-tual physical processess. These “processess” of creationand destruction are nothing but a verbalization of cer-tain mathematical terms appearing only in one particularmethod of calculation – the perturbation expansion withthe square bracket in (93) treated as the perturbation.Last but not least, even if, despite the unnaturalness,one decides to express everything in terms of the opera-tors (91) and the states (94), there still may remain anambiguity in choosing the constant ω. All this demon-strates that, in general, QM is not a theory of “quanta”attributed to the operator N .

B. Particles in perturbative QFT

The analogy between the notion of “quanta” in thefirst-quantized theory of particles and the notion of “par-ticles” in QFT is complete. For example, the QFT analogof (95) is the field operator in the Schrodinger picture

φ(x) =∑

k

akfk(x) + a†kf∗k (x), (96)

which corresponds to (76) with (68) and (69), at fixedt. If fk(x, t) are the plane waves proportional toe−i(ωkt−kx), then the quantum Hamiltonian obtained

from the Lagrangian density (82) turns out to be

H =∑

k

ωk

(

Nk +1

2

)

, (97)

with Nk ≡ a†kak, which is an analog of the first term

in (93). This analogy is related to the fact that (82)represents a relativistic-field generalization of the har-monic oscillator. (The harmonic-oscillator Lagrangian isquadratic in x and its derivative, while (82) is quadraticin φ and its derivatives). The Hamiltonian (97) hasa clear physical interpretation; ignoring the term 1/2(which corresponds to an irrelevant ground-state energy∑

kωk/2), for each ωk there can be only an integer num-

ber nk of quanta with energy ωk, so that their total en-ergy sums up to nkωk. These quanta are naturally in-terpreted as “particles” with energy ωk. However, theLagrangian (82) is only a special case. In general, a La-grangian describing the field φ may have a form

L =1

2(∂µφ)(∂µφ) − V (φ), (98)

where V (φ) is an arbitrary potential. Thus, in gen-eral, the Hamiltonian contains an additional term anal-ogous to that in (93), which destroys the “particle”-interpretation of the spectrum.

Whereas the formal mathematical analogy betweenfirst quantization and QFT (which implies the irrelevance

of the number operator N) is clear, there is one crucialphysical difference: Whereas in first quantization thereis really no reason to attribute a special meaning to theoperator N , there is an experimental evidence that thisis not so for QFT. The existence of particles is an exper-imental fact! Thus, if one wants to describe the exper-imentally observed objects, one must either reject QFT(which, indeed, is what many elementary-particle physi-cists were doing in the early days of elementary-particlephysics and some of them are doing it even today [49]),or try to artificially adapt QFT such that it remains atheory of particles even with general interactions (such asthose in (98)). From a pragmatic and phenomenologicalpoint of view, the latter strategy turns out to be surpris-ingly successful! For example, in the case of (98), oneartificially defines the interaction part of the Lagrangianas

VI(φ) = V (φ) − 1

2m2φ2, (99)

and treats it as a perturbation of the “free” Lagrangian(82). For that purpose, it is convenient to introduce amathematical trick called interaction picture, which is apicture that interpolates between the Heisenberg picture(where the time dependence is attributed to fields φ) andthe Schrodinger picture (where the time dependence isattributed to states |Ψ〉). In the interaction picture, thefield satisfies the free Klein-Gordon equation of motion,while the time evolution of the state is governed only bythe interaction part of the Hamiltonian. This trick allows

19

one to use the free expansion (76) with (68) and (69), de-

spite the fact that the “true” quantum operator φ(x, t) inthe Heisenberg picture cannot be expanded in that way.In fact, all operators in the interaction picture satisfy thefree equations of motion, so the particle-number opera-tor can also be introduced in the same way as for freefields. Analogously to the case of first quantization dis-cussed after Eq. (95), certain mathematical terms in theperturbation expansion can be pictorially represented bythe so-called Feynman diagrams. (For technical details,I refer the reader to [47, 48].) In some cases, the finalmeasurable results obtained in that way turn out to bein excellent agreement with experiments.

C. Virtual particles?

The calculational tool represented by Feynman dia-grams suggests an often abused picture according towhich “real particles interact by exchanging virtual par-ticles”. Many physicists, especially nonexperts, take thispicture literally, as something that really and objectivelyhappens in nature. In fact, I have never seen a populartext on particle physics in which this picture was not pre-sented as something that really happens. Therefore, thispicture of quantum interactions as processes in which vir-tual particles exchange is one of the most abused myths,not only in quantum physics, but in physics in general.Indeed, there is a consensus among experts for founda-tions of QFT that such a picture should not be takenliterally. The fundamental principles of quantum theorydo not even contain a notion of a “virtual” state. The no-tion of a “virtual particle” originates only from a specificmathematical method of calculation, called perturbativeexpansion. In fact, perturbative expansion representedby Feynman diagrams can be introduced even in clas-sical physics [50, 51], but nobody attempts to verbalizethese classical Feynman diagrams in terms of classical“virtual” processes. So why such a verbalization is tol-erated in quantum physics? The main reason is the factthat the standard interpretation of quantum theory doesnot offer a clear “canonical” ontological picture of theactual processes in nature, but only provides the proba-bilities for the final results of measurement outcomes. Inthe absence of such a “canonical” picture, physicists takethe liberty to introduce various auxiliary intuitive pic-tures that sometimes help them think about otherwiseabstract quantum formalism. Such auxiliary pictures, bythemselves, are not a sin. However, a potential problemoccurs when one forgets why such a picture has been in-troduced in the first place and starts to think on it tooliterally.

D. Nonperturbative QFT

In some cases, the picture of particles suggested by the“free” part of the Lagrangian does not really correspond

to particles observed in nature. The best known exam-ple is quantum chromodynamics (QCD), a QFT theorydescribing strong interactions between quarks and glu-ons. In nature we do not observe quarks, but rathermore complicated particles called hadrons (such as pro-tons, neutrons, and pions). In an oversimplified but oftenabused picture, hadrons are built of 2 or 3 quarks gluedtogether by gluons. However, since free quarks are neverobserved in nature, the perturbative expansion, so suc-cessful for some other QFT theories, is not very successfulin the case of QCD. Physicists are forced to develop otherapproximative methods to deal with it. The most suc-cessful such method is the so-called lattice QCD (for anintroductory textbook see [52] and for pedagogic reviewssee [53, 54]). In this method, the spacetime continuumis approximated by a finite lattice of spacetime points,allowing the application of brutal-force numerical meth-ods of computation. This method allows to compute theexpectation values of products of fields in the groundstate, by starting from first principles. However, to ex-tract the information about particles from these purelyfield-theoretic quantities, one must assume a relation be-tween these expectation values and the particle quanti-ties. This relation is not derived from lattice QCD itself,but rather from the known relation between fields andparticles in perturbative QFT. Consequently, althoughthis method reproduces the experimental hadron datamore-or-less successfully, the concept of particle in thismethod is not more clear than that in the perturbativeapproach. Thus, the notion of real particles is not de-rived from first principles and nothing in the formalismsuggests a picture of the “exchange of virtual particles”.

E. Particles and the choice of time

As we have seen, although the notion of particles ininteracting QFT theories cannot be derived from firstprinciples (or at least we do not know yet how to dothat), there are heuristic mathematical procedures thatintroduce the notion of particles that agrees with ex-periments. However, there are circumstances in QFTin which the theoretical notion of particles is even moreambiguous, while present experiments are not yet able toresolve these ambiguities. Consider again the free fieldexpanded as in (76) with (68) and (69). The notion ofparticles rests on a clear separation between the creation

operators a†k and the destruction operators ak. The def-inition of these operators is closely related to the choiceof the complete orthonormal basis fk(x, t) of solutionsto the classical Klein-Gordon equation. However, thereare infinitely many different choices of this basis. Theplane-wave basis

fk(x, t) ∝ e−i(ωkt−kx) ≡ e−ik·x (100)

is only a particular convenient choice. Different choicesmay lead to different creation and destruction operators

a†k and ak, and thus to different notions of particles. How

20

to know which choice is the right one? Eq. (100) sug-gests a physical criterion according to which the modesfk(x, t) should be chosen such that they have a positivefrequency. However, the notion of frequency assumes thenotion of time. On the other hand, according to the the-ory of relativity, there is not a unique choice of the timecoordinate. Therefore, the problem of the right defini-tion of particles reduces to the problem of the right defi-nition of time. Fortunately, the last exponential functionin (100) shows that the standard plane waves fk(x) areLorentz invariant, so that different time coordinates re-lated by a Lorentz transformation lead to the same def-inition of particles. However, Lorentz transformationsrelate only proper coordinates attributed to inertial ob-servers in flat spacetime. The general theory of relativ-ity allows much more general coordinate transformations,such as those that relate an inertial observer with an ac-celerating one. (For readers who are not familiar withgeneral theory of relativity there are many excellent in-troductory textbooks, but my favored one that I highlyrecommend to the beginners is [55]. For an explicit con-struction of the coordinate transformations between aninertial observer and an arbitrarily moving one in flatspacetime, see [56, 57], and for instructive applications,see [57, 58, 59]. Nevertheless, to make this paper read-able by those who are not familiar with general relativ-ity, in the rest of Sec. IX, as well as in Sec. X, I omitsome technical details that require a better understand-ing of general relativity, keeping only the details thatare really necessary to understand the quantum aspectsthemselves.) Different choices of time lead to differentchoices of the positive-frequency bases fk(x), and thus

to different creation and destruction operators a†k and ak,respectively. If

φ(x) =∑

k

akfk(x) + a†kf∗k (x),

φ(x) =∑

l

ˆalfl(x) + ˆa†l f

∗l (x) (101)

are two such expansions in the bases fk(x) and fl(x),respectively, it is easy to show that the correspondingcreation and destruction operators are related by a lineartransformation

ˆal =∑

k

αlkak + β∗lk a

†k,

ˆa†l =

∑

k

α∗lk a

†k + βlkak, (102)

where

αlk ≡ (fl, fk), β∗lk ≡ (fl, f

∗k ), (103)

are given by the scalar products defined as in (59). (Toderive (102), take the scalar product of both expressionsin (101) with fl′ on the left and use the orthonormalityrelations (fl′ , fl) = δl′l, (fl′ , f

∗l ) = 0.) The transforma-

tion (102) between the two sets of creation and destruc-tion operators is called Bogoliubov transformation. Since

both bases are orthonormal, the Bogoliubov coefficients(103) satisfy

∑

k

(αlkα∗l′k − β∗

lkβl′k) = δll′ , (104)

where the negative sign is a consequence of the fact thatnegative frequency solutions have negative norms, i.e.,(f∗

k , f∗k′) = −δkk′ , (f∗

l , f∗l′) = −δll′ . One can show that

(104) provides that ˆal and ˆa†l also satisfy the same com-

mutation relations (70) as ak and a†k do. A physicallynontrivial Bogoliubov transformation is that in which atleast some of the βlk coefficients are not zero. Two dif-ferent definitions of the particle-number operators are

N =∑

k

Nk,ˆN =

∑

l

ˆN l, (105)

where

Nk = a†kak,ˆN l = ˆa

†lˆal. (106)

In particular, from (102), it is easy to show that the vac-

uum |0〉 having the property N |0〉 = 0 has the property

〈0| ˆN l|0〉 =∑

k

|βlk|2. (107)

For a nontrivial Bogoliubov transformation, this meansthat the average number of particles in the no-particlestate |0〉 is a state full of particles when the particles are

defined by ˆN instead of N ! Conversely, the no-particle

state |0〉 having the property ˆN |0〉 = 0 is a state full

of particles when the particles are defined with N . So,what is the right operator of the number of particles, N

or ˆN? How to find the right operator of the number ofparticles? The fact is that, in general, a clear universallyaccepted answer to this question is not known! Instead,there are several possibilities that we discuss below.

One possibility is that the dependence of the particleconcept on the choice of time means that the concept ofparticles depends on the observer. The best known exam-ple of this interpretation is the Unruh effect [60, 61, 62],according to which a uniformly accelerating observer per-ceives the standard Minkowski vacuum (defined with re-spect to time of an inertial observer in Minkowski flatspacetime) as a state with a huge number of particleswith a thermal distribution of particle energies, with thetemperature proportional to the acceleration. Indeed,this effect (not yet experimentally confirmed!) can beobtained by two independent approaches. The first ap-proach is by a Bogoliubov transformation as indicatedabove, leading to [60, 62]

〈0| ˆN l|0〉 =1

e2πωl/a − 1, (108)

where a is the proper acceleration perceived by the accel-erating observer, ωl is the frequency associated with the

21

solution fl(x), and we use units in which h = c = 1. (Co-ordinates of a uniformly accelerating observer are knownas Rindler coordinates [63], so the quantization based onparticles defined with respect to the Rindler time is calledRindler quantization.) We see that the right-hand sideof (108) looks just as a Bose-Einstein distribution at thetemperature T = a/2π (in units in which the Boltzmannconstant is also taken to be unit). The second approachis by studying the response of a theoretical model of anaccelerating particle detector, using only the standardMinkowski quantization without the Bogoliubov trans-formation. However, these two approaches are not equiv-alent [64, 65]. Besides, such a dependence of particles onthe observer is not relativistically covariant. In partic-ular, it is not clear which of the definitions of particles,if any, acts as a source for a (covariantly transforming)gravitational field.

An alternative is to describe particles in a unique co-variant way in terms of local particle currents [66], butsuch an approach requires a unique choice of a preferredtime coordinate. For example, for a hermitian scalar field(76), the particle current is

jPµ (x) = iψ†(x)↔∂µ ψ(x), (109)

which (unlike (55) with (60)) requires the identification

of the positive- and negative-frequency parts ψ(x) and

ψ†(x), respectively. Noting that the quantization of fieldsthemselves based on the functional Schrodinger equation(87) also requires a choice of a preferred time coordinate,it is possible that a preferred time coordinate emergesdynamically from some nonstandard covariant methodof quantization, such as that in [38].

Another possibility is that the concept of particles asfundamental objects simply does not make sense in QFT[62, 67]. Instead, all observables should be expressed interms of local fields that do not require the artificial iden-tification of the positive- and negative-frequency parts.For example, such an observable is the Hamiltonian den-sity (84), which represents the T 0

0 -component of the co-variant energy-momentum tensor T µ

ν (x). Whereas suchan approach is very natural from the theoretical point ofview according to which QFT is nothing but a quantumtheory of fields, the problem is to reconcile it with the factthat the objects observed in high-energy experiments are– particles.

F. Particle creation by a classical field

When the classical metric gµν(x) has a nontrivial de-pendence on x, the Klein-Gordon equation (53) for thefield φ(x) generalizes to

(

1√

|g|∂µ

√

|g|gµν∂ν +m2

)

φ = 0, (110)

where g is the determinant of the matrix gµν . In par-ticular, if the metric is time dependent, then a solution

fk(x) having a positive frequency at some initial time tinmay behave as a superposition of positive- and negative-frequency solutions at some final time tfin. At the finaltime, the solutions that behave as positive-frequency onesare some other solutions fl(x). In this case, it seems

natural to define particles with the operator N at the

initial time and with ˆN at the final time. If the time-independent state in the Heisenberg picture is given bythe “vacuum” |0〉, then 〈0|N |0〉 = 0 denotes that thereare no particles at tin, while (107) can be interpreted as aconsequence of an evolution of the particle-number oper-ator, so that (107) refers only to tfin. This is the essence ofthe mechanism of particle creation by a classical gravita-tional field. The best known example is particle creationby a collapse of a black hole, known also as Hawking radi-ation [68]. (For more details, see also the classic textbook[62], a review [69], and a pedagogic review [70].) Simi-larly to the Unruh effect (108), the Hawking particleshave the distribution

〈0| ˆN l|0〉 =1

e8πGMωl − 1, (111)

whereG is the Newton gravitational constant which has adimension (energy)−2 andM is the black-hole mass. Thisis the result obtained by defining particles with respectto a specific time, that is, the time of an observer staticwith respect to the black hole and staying far from theblack-hole horizon. Although (111) looks exactly like aquantum Bose-Einstein thermal distribution at the tem-perature

T =1

8πGM, (112)

this distribution is independent of the validity of thebosonic quantum commutation relations (70). Instead, itturns out that the crucial ingredient leading to a thermaldistribution is the existence of the horizon [71], which isa classical observer-dependent general-relativistic objectexisting not only for black holes, but also for accelerat-ing observers in flat spacetime. Thus, the origin of thisthermal distribution can be understood even with clas-sical physics [72], while only the mechanism of particlecreation itself is intrinsically quantum. In the literature,the existence of thermal Hawking radiation often seemsto be widely accepted as a fact. Nevertheless, since itis not yet experimentally confirmed and since it rests onthe theoretically ambiguous concept of particles in curvedspacetime (the dependence on the choice of time), cer-tain doubts on its existence are still reasonable (see, e.g.,[66, 73, 74] and references therein). Thus, the existenceof Hawking radiation can also be qualified as a myth.

The classical gravitational field is not the only classicalfield that seems to be able to cause a production of parti-cles from the vacuum. The classical electric field seems tobe able to produce particle-antiparticle pairs [69, 75, 76],by a mechanism similar to the gravitational one. Fordiscussions of theoretical ambiguities lying behind thistheoretically predicted effect, see [66, 73, 77].

22

G. Particles, fields, or something else?

Having in mind all these foundational problems withthe concept of particle in QFT, it is still impossible toclearly and definitely answer the question whether theworld is made of particles or fields. Nevertheless, practi-cally oriented physicists may not find this question dis-turbing as long as the formalism, no matter how inco-herent it may appear to be, gives correct predictions onmeasurable quantities. Such a practical attitude seemsto be justified by the vagueness of the concept of re-ality inherent to QM itself. Indeed, one can adopt ahard version of orthodox interpretation of QM accordingto which information about reality is more fundamentalthan reality itself and use it to justify the noncovariantdependence of particles (as well as some other quantities)on the observer [78]. However, in the standard orthodoxQM, where rigorous no-hidden-variable theorems exist(see Sec. V), at least the operators are defined unam-biguously. Thus, even the hard-orthodox interpretationof QM is not sufficient to justify the interpretation ofthe particle-number-operator ambiguities as different re-alities perceived by different observers. An alternative tothis orthodox approach is an objective-realism approachin which both particles and fields separately exist, whichis a picture that seems to be particularly coherent in theBohmian interpretation [79].

Finally, there is a possibility that the world is madeneither of particles nor of fields, but of strings. (Foran excellent pedagogic introduction to string theory see[80]. In particular, this book also breaks one myth inphysics – the myth that string theory is mathematicallyan extremely complicated theory that most other physi-cists cannot easily understand. For a more concise ped-agogic introduction to string theory see also [81].) Infact, many string theorists speak about the existence ofstrings as a definite fact. Fortunately, there is still a suf-ficiently large number of authoritative physicists that arehighly skeptical about string theory, which does not allowstring theory to become a widely accepted myth. Nev-ertheless, string theory possesses some remarkable the-oretical properties that makes it a promising candidatefor a more fundamental description of nature. Accord-ing to this theory, particles are not really pointlike, butextended one-dimensional objects. Their characteristiclength, however, is very short, which is why they appearas pointlike with respect to our current experimental abil-ities to probe short distances. However, just as for parti-cles, there is first quantization of strings, as well as secondquantization that leads to string field theory. Thus, evenif string theory is correct, there is still a question whetherthe fundamental objects are strings or string fields. How-ever, while first quantization of strings is well understood,string field theory is not. Moreover, there are indicationsthat string field theory may not be the correct approachto treat strings [82]. Consequently, particles (that rep-resent an approximation of strings) may be more fun-damental than fields. Concerning the issue of objective

reality, there are indications that the Bohmian interpre-tation of strings may be even more natural than that ofparticles [83]. The Bohmian interpretation of strings alsobreaks some other myths inherent to string theory [84].

X. BLACK-HOLE ENTROPY ISPROPORTIONAL TO ITS SURFACE

As this claim is not yet a part of standard textbooks,this is not yet a true myth. Nevertheless, in the last10 or 20 years this claim has been so often repeated byexperts in the field (the claim itself is about 30 yearsold) that it is very likely that it will soon become a truemyth. Before it happens, let me warn the wider physicscommunity that this claim is actually very dubious.

A. Black-hole “entropy” in classical gravity

The claim in the title of this section is actually a partof a more general belief that there exists a deep relationbetween black holes and thermodynamics. The first ev-idence supporting this belief came from certain classicalproperties of black holes that, on the mathematical level,resemble the laws of thermodynamics [85, 86]. (For gen-eral pedagogic overviews, see, e.g., [55, 87] and for anadvanced pedagogic review with many technical details,see [88].) Black holes are dynamical objects that canstart their evolution from a huge number of different ini-tial states, but eventually end up in a highly-symmetricequilibrium stationary state specified only by a few globalconserved physical quantities, such as their mass M (i.e.,energy E), electric charge Q, and angular momentum J .The physical laws governing the behavior of such equi-librium black holes formally resemble the laws governingthe behavior of systems in thermodynamic equilibrium.The well-known four laws of thermodynamics have thefollowing black-hole analogues:

• Zeroth law: There exists a local quantity called sur-face gravity κ (which can be viewed as the general-relativistic analog of the Newton gravitational fieldGM/r2) that, in equilibrium, turns out to be con-stant everywhere on the black-hole horizon. Thisis an analog of temperature T which is constant inthermodynamic equilibrium.

• First law: This is essentially the law of energy con-servation, which, both in the black-hole and thethermodynamic case, has an origin in even morefundamental laws. As such, this analogy shouldnot be surprising, but it is interesting that in bothcases the conservation of energy takes a mathemat-ically similar form. For black holes it reads

dM =κ

8πGdA+ ΩdJ + ΦdQ, (113)

where A is the surface of the horizon, Ω is the angu-lar velocity of the horizon, and Φ is the electrostatic

23

potential at the horizon. This is analogous to thethermodynamic first law

dE = TdS − pdV + µdN, (114)

where S is the entropy, p is the pressure, V is thevolume, µ is the chemical potential, and N is thenumber of particles. In particular, note that theblack-hole analog of the entropy S is a quantityproportional to the black-hole surface A. This al-lows us to introduce the black-hole “entropy”

Sbh = αA, (115)

where α is an unspecified constant.

• Second law: Although the fundamental microscopicphysical laws are time reversible, the macroscopiclaws are not. Instead, disorder tends to increasewith time. In the thermodynamic case, it meansthat entropy cannot decrease with time, i.e., dS ≥0. In the gravitational case, owing to the attractivenature of the gravitational force, it turns out thatthe black-hole surface cannot decrease with time,i.e., dA ≥ 0.

• Third law: It turns out that, by a realistic physi-cal process, it is impossible to reach the state withκ = 0. This is analogous to the third law of thermo-dynamics according to which, by a realistic physi-cal process, it is impossible to reach the state withT = 0.

Although the analogy as presented above is suggestive,it is clear that classical black-hole parameters are con-ceptually very different from the corresponding thermo-dynamic parameters. Indeed, the formal analogies abovewere not taken very seriously at the beginning. In partic-ular, an ingredient that is missing for a full analogy be-tween classical black holes and thermodynamic systemsis – radiation with a thermal spectrum. Classical blackholes (i.e., black holes described by the classical Einsteinequation of gravity) do not produce radiation with a ther-mal spectrum.

B. Black-hole “entropy” in semiclassical gravity

A true surprise happened when Hawking found out [68]that semiclassical (i.e., gravity is treated classically whilematter is quantized) black holes not only radiate (which,by itself, is not a big surprise), but radiate exactly witha thermal spectrum at a temperature proportional to κ.In the special case of a black hole with J = Q = 0, thistemperature is equal to (112). Since dJ = dQ = 0, weattempt to write (113) as

dSbh =dM

T, (116)

which corresponds to (114) with dV = dN = 0. From(112), we see that

dM

T= 8πGMdM. (117)

From the Schwarzschild form of the black-hole metric inthe polar spacial coordinates (r, ϑ, ϕ) (see, e.g., [55])

ds2 =dt2

1 − 2GM

r

−(

1 − 2GM

r

)

dr2−r2(dϑ2+sin2ϑ dϕ2),

(118)we see that the horizon corresponding to the singularbehavior of the metric is at the radius

r = 2GM. (119)

Consequently, the surface of the horizon is equal to

A = 4πr2 = 16πG2M2. (120)

Therefore, (115) implies

dSbh = α32πG2MdM. (121)

Thus, we see that (117) and (121) are really consistentwith (116), provided that α = 1/4G. Therefore, (115)becomes

Sbh =A

4G. (122)

In fact, (122) turns out to be a generally valid relation,for arbitrary J and Q.

Now, with the results (112) and (122), the analogybetween black holes and thermodynamics seems to bemore complete. Nevertheless, it is still only an analogy.Moreover, thermal radiation (which is a kinematical ef-fect depending only on the metric) is not directly logicallyrelated to the four laws of classical black-hole “thermody-namics” (for which the validity of the dynamical Einsteinequations is crucial) [89]. Still, many physicists believethat such a striking analogy cannot be a pure formal co-incidence. Instead, they believe that there is some evendeeper meaning of this analogy. In particular, as classicalhorizons hide information from observers, while the or-thodox interpretation of QM suggests a fundamental roleof information available to observers, it is believed thatthis could be a key to a deeper understanding of the rela-tion between relativity and quantum theory [78]. As thecorrect theory of quantum gravity is not yet known (forreviews of various approaches to quantum gravity, see[90, 91]), there is a belief that this deeper meaning willbe revealed one day when we better understand quantumgravity. Although this belief may turn out to be true, atthe moment there is no real proof that this necessarilymust be so.

A part of this belief is that (122) is not merely a quan-tity analogous to entropy, but that it really is the en-tropy. However, in standard statistical physics (from

24

which thermodynamics can be derived), entropy is aquantity proportional to the number of the microscopicphysical degrees of freedom. On the other hand, thederivation of (122) as sketched above does not providea direct answer to the question what, if anything, thesemicroscopic degrees of freedom are. In particular, theycannot be simply the particles forming the black hole, asthere is no reason why the number of particles should beproportional to the surface A of the black-hole boundary.Indeed, as entropy is an extensive quantity, one expectsthat it should be proportional to the black-hole volume,rather than to its surface. It is believed that quantumgravity will provide a more fundamental answer to thequestion why the black-hole entropy is proportional toits surface, rather than to its volume. Thus, the pro-gram of finding a microscopic derivation of Eq. (122) issometimes referred to as “holly grail” of quantum gravity.(The expression “holly grail” fits nice with my expression“myth”.)

C. Other approaches to black-hole entropy

Some results in quantum gravity already suggest a mi-croscopic explanation of the proportionality of the black-hole entropy with its surface. For example, a loop rep-resentation of quantum-gravity kinematics (for reviews,see, e.g., [92, 93]) leads to a finite value of the entropyof a surface, which coincides with (122) if one additionalfree parameter of the theory is adjusted appropriately.However, loop quantum gravity does not provide a newanswer to the question why the black-hole entropy shouldcoincide with the entropy of its boundary. Instead, ituses a classical argument for this, based on the observa-tion that the degrees of freedom behind the horizon areinvisible to outside observers, so that only the bound-ary of the black hole is relevant to physics observed byoutside observers. (The book [93] contains a nice peda-gogic presentation of this classical argument. Besides,it contains an excellent pedagogic presentation of therelational interpretation of general relativity, which, inparticular, may serve as a motivation for the conceptu-ally much more dubious relational interpretation of QM[25, 26] mentioned in Sec. V.) Such an explanation of theblack-hole entropy is not what is really searched for, as itdoes not completely support the four laws of black-hole“thermodynamics”, since the other extensive quantitiessuch as mass M and charge Q contain information aboutthe matter content of the interior. What one wants toobtain is that the entropy of the interior degrees of free-dom is proportional to the boundary of the interior.

A theory that is closer to achieving this goal is stringtheory, which, among other things, also contains a quan-tum theory of gravity. Strings are one-dimensional ob-jects containing an infinite number of degrees of freedom.However, not all degrees of freedom need to be excited. Inlow-energy states of strings, only a few degrees of freedomare excited, which corresponds to states that we perceive

as standard particles. However, if the black-hole interiorconsists of one or a few self-gravitating strings in highlyexcited states, then the entropy associated with the mi-croscopic string degrees of freedom is of the order ofGM2

(for reviews, see [80, 94]). This coincides with the semi-classical black-hole “entropy”, as the latter is also of theorder of GM2, which can be seen from (122) and (120).The problem is that strings do not necessarily need to bein highly excited states, so the entropy of strings does notneed to be of the order of GM2. Indeed, the black-holeinterior may certainly contain a huge number of stan-dard particles, which corresponds to a huge number ofstrings in low-excited states. It is not clear why the en-tropy should be proportional to the black-hole surfaceeven then.

A possible reinterpretation of the relation (122) is thatit does not necessarily denote the actual value of theblack-hole entropy, but only the upper limit of it. Thisidea evolved into a modern paradigm called holographicprinciple (see [95] for a review), according to which theboundary of a region of space contains a lot of infor-mation about the region itself. However, a clear gen-eral physical explanation of the conjectured holographicprinciple, or that of the conjectured upper limit on theentropy in a region, is still missing.

Finally, let me mention that the famous black-hole en-tropy paradox that seems to suggest the destruction ofentropy owing to the black-hole radiation (for pedagogicreviews, see [96, 97]) is much easier to solve when Sbh

is not interpreted as true entropy [98]. Nevertheless, Iwill not further discuss it here as this paper is not aboutquantum paradoxes (see, e.g., [16]), but about quantummyths.

XI. DISCUSSION AND CONCLUSION

As we have seen, QM is full of “myths”, that is, claimsthat are often presented as definite facts, despite the factthat the existing evidence supporting these claims is notsufficient to proclaim them as true facts. To show thatthey are not true facts, I have also discussed the draw-backs of this evidence, as well as some alternatives. In thepaper, I have certainly not mentioned all myths existingin QM, but I hope that I have catched the most famousand most fundamental ones, appearing in several funda-mental branches of physics ranging from nonrelativisticquantum mechanics of single particles to quantum grav-ity and string theory.

The question that I attempt to answer now is – why themyths in QM are so numerous? Of course, one of the rea-sons is certainly the fact that we still do not completelyunderstand QM at the most fundamental level. However,this fact by itself does not explain why quantum physi-cists (who are supposed to be exact scientists) are so tol-erant and sloppy about arguments that are not really theproofs, thus allowing the myths to form. To find a deeperreason, let me first note that the results collected and re-

25

viewed in this paper show that the source of disagreementamong physicists on the validity of various myths is not ofmathematical origin, but of conceptual one. However, inclassical mechanics, which is well-understood not only onthe mathematical, but also on the conceptual level, sim-ilar disagreement among physicists almost never occur.Thus, the common origin of myths in QM must lie in thefundamental conceptual difference between classical andquantum mechanics. But the main conceptual differencebetween classical and quantum mechanics that makes thelatter less understood on the conceptual level is the factthat the former introduces a clear notion of objective re-ality even without measurements. (This is why I referredto the myth of Sec. V as the central myth in QM.) Thus, Iconclude that the main reason for the existence of mythsin QM is the fact that QM does not give a clear answerto the question what, if anything, objective reality is.

To support the conclusion above, let me illustrate itby a simple model of objective reality. Such a modelmay seem to be naive and unrealistic, or may be opento further refinements, but here its only purpose is todemonstrate how a model with explicit objective real-ity immediately gives clear unambiguous answers to thequestions whether the myths discussed in this paper aretrue or not. The simple model of objective reality I dis-cuss is a Bohmian-particle interpretation, according towhich particles are objectively existing pointlike objectshaving deterministic trajectories guided by (also objec-tively existing) wave functions. To make the notion ofparticles and their “instantaneous” interactions at a dis-tance unique, I assume that there is a single preferred sys-tem of relativistic coordinates, roughly coinciding withthe global system of coordinates with respect to whichthe cosmic microwave backround is homogeneous andisotropic. Now let me briefly consider the basic claimsof the titles of all sections of the paper. Is there a wave-particle duality? Yes, because both particles and wavefunctions objectively exist. Is there a time-energy un-certainty relation? No, at least not at the fundamentallevel, because the theory is deterministic. Is nature fun-damentally random? No, in the sense that both wavesand particle trajectories satisfy deterministic equations.Is there reality besides the measured reality? Yes, by thecentral assumption of the model. Is QM local or non-local? It is nonlocal, as it is a hidden-variable theoryconsistent with standard statistical predictions of QM.Is there a well-defined relativistic QM? Yes, because, byassumption, relativistic particle trajectories are well de-fined with the aid of a preferred system of coordinates.Does quantum field theory (QFT) solves the problems ofrelativistic QM? No, because particles are not less funda-mental than fields. Is QFT a theory of particles? Yes, be-cause, by assumption, particles are fundamental objects.(If the current version of QFT is not completely compat-ible with the fundamental notion of particles, then it isQFT that needs to be modified.) Is black-hole entropyproportional to its surface? To obtain a definite answerto this last question, I have to further specify my model of

objective reality. For simplicity, I assume that gravity isnot quantized (currently known facts do not actually ex-clude this possibility), but determined by a classical-likeequation that, at least at sufficiently large distances, hasthe form of a classical Einstein equation in which “mat-ter” is determined by the actual particle positions andvelocities. In such a model, the four laws of black-hole“thermodynamics” are a direct consequence of the Ein-stein equation, and there is nothing to be explained aboutthat. The quantity Sbh is only analogous to entropy, sothe answer to the last question is – no. Whatever (if any-thing) the true quantum mechanism of objective particlecreation near the black-hole horizon might be (owing tothe existence of a preferred time, the mechanism based onthe Bogoliubov transformation seems viable), the classi-cal properties of gravity near the horizon imply that thedistribution of particle energies will be thermal far fromthe horizon, which also does not require an additionalexplanation and is not directly related to the four lawsof black-hole thermodynamics [89].

Of course, with a different model of objective reality,the answers to some of the questions above may be differ-ent. But the point is that the answers are immediate andobvious. With a clear notion of objective reality, there isnot much room for myths and speculations. It does notprove that objective reality exists, but suggests that thisis a possibility that should be considered more seriously.

To conclude, the claim that the fundamental principlesof quantum theory are today completely understood, sothat it only remains to apply these principles to variouspractical physical problems – is also a myth. Instead,quantum theory is a theory which is not yet completelyunderstood at the most fundamental level and is opento further fundamental research. Through this paper, Ihave demonstrated this by discussing various fundamen-tal myths in QM for which a true proof does not yet reallyexist. I have also demonstrated that all these myths are,in one way or another, related to the central myth in QMaccording to which objective unmeasured reality does notexist. I hope that this review will contribute to a bettergeneral conceptual understanding of quantum theory andmake readers more cautios and critical before acceptingvarious claims on QM as definite facts.

Acknowledgments

As this work comprises the foundational backgroundfor a large part of my own scientific research in sev-eral seemingly different branches of theoretical physics,it is impossible to name all my colleagues specialized indifferent branches of physics that indirectly influencedthis work through numerous discussions and objectionsthat, in particular, helped me become more open mindedby understanding how the known physical facts can beviewed and interpreted in many different inequivalentways, without contradicting the facts themselves. Thiswork was supported by the Ministry of Science and Tech-

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