DEVELOPMENT OF THE THEORY OF ANTIPARTICLES, CHARGES OF ELEMENTARY PARTICLES, AND PROPERTIES OF HEAVY NEUTRAL MESONS

Ya. B. Zel’dovich

1. PARTICLES AND ANTIPARTICLES

The existence, predicted by Dirac, of the positron—the antiparticle of the electron—has forever entered the history of physics as a major achievement of theory.

The path toward a correct understanding of the relation between particles and antiparticles was not an easy one. The Dirac equation, which correctly described the properties of the electron, was established in 1929. This equation, satisfying the requirements of the theory of relativity, led to the conclusion that the electron can be found both in states with positive energy and in states with negative energy: in the theory of relativity

\[ E^2=(m_0c^2)^2+p^2c^2 \]

(\(E\) is energy, \(m_0\) the rest mass, \(p\) the momentum, \(c\) the speed of light), and extracting the square root gives two signs,

\[ E=\pm \sqrt{(m_0c^2)^2+p^2c^2}. \]

For the first several years, attempts continued to get rid of the states with negative energy, to forbid them; one may point to Schrödinger as the author of one such attempt. However, as I. E. Tamm showed, without levels of negative energy Dirac’s theory is manifestly incomplete and does not describe such important phenomena as the scattering of light by free electrons (the Compton effect).

Dirac took the second step by introducing the idea that, in the normal state, all levels with negative energy are filled. Owing to the Pauli principle, this means that free electrons with \(E>0\) cannot fall into the occupied states with \(E<0\).

The absence of one electron among the states with negative energy (a “hole”) is perceived experimentally as a unit positive charge.

Initially Dirac assumed that in this way one could describe the proton. However, calculations showed that such a view inevitably leads, in the hydrogen atom, to the falling of the electron into a state with negative energy.

Soon after this the positron was discovered experimentally, and the process of formation of \(e^{+}+e^{-}\) pairs and the reverse process of annihilation of positrons with electrons were studied in detail. Let us note in passing that certain finer details of the process (the different probability of annihilation and the different number of gamma quanta formed for parallel and antiparallel spins of the electron and positron) were clarified only comparatively recently, in 1947–1949, in the works of I. Ya. Pomeranchuk\(^{1}\), L. D. Landau\(^{2}\), and E. M. Lifshitz\(^{3}\) (see also\(^{4}\)).

With the discovery of the positron, a complete symmetry between particles and antiparticles was revealed. It became clear that the representation of electrons as “particles in the proper sense of the word” and positrons as “holes” in a background filled with electrons is conventional. With equal success one may call the positron a particle and the electron a hole in a background filled with positrons.

Thus there crystallized the idea of the symmetry of all laws of nature with respect to the two signs of electric charge. The concept arose of charge conjugation—a mental transformation under which all particles are transformed into antiparticles and antiparticles into particles, so that all electric charges and magnetic moments, as well as electromagnetic fields, change sign.

In this case the equations describing the motion of a system (in any approximation) in classical, quantum, and relativistic theory must remain unchanged; they must be invariant with respect to charge conjugation. The question of charge conjugation is treated in detail and rigorously in the pages of Uspekhi Fizicheskikh Nauk in an article by I. S. Shapiro\(^{5}\).

The world around us is clearly not charge-symmetric: the matter around us contains an enormous number of electrons, whereas positrons are formed only under very special conditions—in cosmic rays, in phenomena connected with radioactive decay, or in accelerators. Under ordinary conditions, for example in air, the electron is stable, while the positron annihilates after a short time. It should be emphasized, however, that this is due to the asymmetry of the initial conditions: air contains electrons, but not positrons. In a vacuum, where the initial conditions too are symmetric, electrons and positrons are equally stable, in full accord with the symmetry of the equations.

The next important step was taken by Pauli and Weisskopf in 1934. They developed the relativistic theory of charged particles

with spin 0. The theory of Pauli and Weisskopf describes, for example, \(\pi^+\) and \(\pi^-\) mesons. In doing so, a great similarity was found in the behavior of particles with spin 0 (\(\pi^+\) mesons) and with spin \(1/2\) (electrons and positrons) in electromagnetic fields. The formulae for the creation of \(\pi^+—\pi^-\) pairs by a gamma quantum, the formulae for the probability of annihilation of a \(\pi^+—\pi^-\) pair with the formation of two \(\gamma\)-quanta, etc., differ from the formulae for electrons and positrons only by insignificant numerical factors.

But particles with spin 0 obey Bose statistics (“bosons”), not Fermi statistics; the Pauli principle is not applicable to them! Consequently, the representations of a filled background and holes are not necessary in order to explain the existence of particles and antiparticles with the typical phenomena of pair creation and their annihilation. The “background” and “holes,” from the modern point of view, are scaffolding that played its historical role in the process of erecting the building and have now become unnecessary. They only hinder the perception of the harmony of the completed building and must be removed at the present time.

Initially the theory was constructed on the model of the quantum mechanics of one particle; in elementary quantum mechanics the motion of a particle, the transition of a particle from one state to another, is considered. Therefore the theory of pair creation was at first also constructed as a theory of the transition of a particle from one state (with negative energy) to another state, with positive energy.

Following the quantum mechanics of a particle, a theory of systems with a variable number of particles was developed; such a theory is necessary, in particular, for a rigorous description of the emission and absorption of light, i.e. the creation and disappearance of light quanta. To this theory of so-called second quantization, a great contribution was made by the works of V. A. Fock. The theory of second quantization was also applied to such particles as electrons and positrons.

In the modern exposition of the theory, states with negative electron energy are not considered; instead, the existence of the positron—a particle with charge conjugate to that of the electron—is introduced at once. In all expressions where the appearance of an electron figures, the destruction of a positron is introduced alongside it.

Let us try to explain the character of the formulae without resorting to mathematical calculations.

In the nonrelativistic theory, the motion of an electron is characterized by the matrix element

\[ M = A \times (\text{annihilation of an electron in the initial state}) \times (\text{creation of an electron in the final state}), \tag{1} \]

and only states with positive energy enter here (\(A\) depends on the external field and is proportional to the charge of the electron).

In Dirac’s theory, \(M\) is naturally generalized as

\[ \begin{aligned} M=A\times&(\text{annihilation of an electron in the initial state}\\ &\text{with positive or negative energy})\times(\text{creation}\\ &\text{of an electron in the final state with positive}\\ &\text{or negative energy}). \end{aligned} \tag{II} \]

Finally, in modern notation,

\[ \begin{aligned} M=A\times&(\text{annihilation of an electron in the initial state}\\ &\text{or creation of a positron in the final state})\times(\text{creation}\\ &\text{of an electron in the final state or annihilation}\\ &\text{of a positron in the initial state}), \end{aligned} \tag{III} \]

where again only states with positive energy are considered. It is easy to verify that in Dirac’s theory (II) and in the modern notation (III), by choosing in each bracket one of the two possibilities, we obtain a total of 4 variants, describing the following four physical processes:

1) motion of an electron,
2) annihilation of the pair \(e^+—e^-\),
3) creation of the pair \(e^+—e^-\),
4) motion of a positron.

If the properties of the electron are known, then the quantity \(A\) is known; consequently, the theory also predicts all the properties of the positron.

When applying Dirac’s theory (II), even when considering a single electron, it is necessary to keep in mind the Pauli principle and the filled “background.” In the modern notation there is no need for this. Only electrons and positrons with positive energy are considered. The theory is now based on the principle of charge symmetry (the symmetry of the laws of nature with respect to charge conjugation). In such a theory all the observed facts concerning the scattering of light, the creation and annihilation of pairs, etc., are obtained in complete agreement with experiment (and with the previous Dirac theory) without introducing the artificial concepts of negative energy, background, and holes. In the new theory the complete analogy between fermions (for example, \(e^+\), \(e^-\)) and bosons (for example, \(\pi^+\), \(\pi^-\)) is transparently clear.

The conceptual development in this field was essentially completed 20 years ago. It is natural to ask whether, in the pages of UFN, it is appropriate to recall such results, which have become classical. Substantially new theoretical conclusions concerning heavy neutral mesons will be set forth only in the second part of the article.

But in the field of the classical theory of particles and antiparticles, too, a number of important experimental results have been obtained in recent years, indicating that this field continues to attract interest, and also demonstrating the fruitfulness of the principle of charge symmetry.

This includes, first of all, the experimental proof of the exact equality of the masses of the electron and the positron. After the result obtained in a 1951 paper[^6] contradicted the symmetry \((m_{e+}=0.998\,m_{e-})\), more accurate measurements in 1953[^7] restored the truth \((m_{e+}-m_{e-}=(-26\pm 71)\,10^{-6}m_e)\). Charge symmetry is confirmed with all experimentally attainable accuracy for mesons as well. We give the latest data on masses and lifetimes[^8][^9][^10][^47]:

The most important result of 1955 was the discovery of the antiproton \(\bar p\) by Segrè’s group at the 6-BeV accelerator[^11]. The existence of the antiproton \(\bar p\) had been predicted long before; it is so obvious and inevitable a consequence of the principle of charge symmetry that it is difficult even to establish by whom and when this prediction was first made.

A somewhat indirect, but nevertheless concrete, confirmation of the concept of antinucleons was the decay of \(\pi^0\). Let us recall that the existence of the neutral \(\pi\)-meson, \(\pi^0\), was predicted by Kemmer on the basis of the charge independence of nuclear forces; a theory in which only the charged \(\pi^+\) and \(\pi^-\) mesons appear does not agree with experiment (see Bethe, UFN[^12]).

It seemed that the neutral \(\pi^0\) should not interact with the electromagnetic field. However, Oppenheimer noted that the \(\pi^0\) can virtually (with a temporary violation of the law of conservation of energy) produce a proton–antiproton pair capable of annihilating with the emission of two gamma quanta. Thus, with the aid of the concept of the antiproton, the decay \(\pi^0=2\gamma\) was predicted and was soon found experimentally.

Such indirect confirmation of the existence of the antiproton was still not sufficiently convincing. The interval of time between the prediction of the antiproton and its observation in 1955 was too great, and some theorists’ nerves did not hold out—in recent years attempts appeared to construct a theory without the antiproton. Now the theory of charge symmetry has triumphed completely.

The fact that, before the start-up of a powerful accelerator, it had not been possible to observe antiprotons \(\bar p\) in cosmic rays is naturally explained by the low probability of formation of \(\bar p\) even in those collisions,

in which the energy of the incident particle is sufficiently large. The point is that, along with the birth of a \(\bar p - p\) pair, in these collisions there competes the more probable production of a large number of \(\pi\)-mesons. According to Fermi’s statistical theory, in collisions whose energy is many times greater than the energy necessary for the production of one pair, the number of antinucleons formed at the moment of collision becomes of the same order as the number of mesons. However, as Pomeranchuk has remarked, in such collisions the interaction continues during the expansion and cooling of the cluster of mesons, nucleons, and antinucleons formed at the very instant of collision. By the time the interaction ceases, when that final composition of the cluster is reached which will be observed by the experimenter, the probability that the antiproton will survive is small. The statistical theory improved by Soviet scientists is set forth in detail in the pages of UFN[^13]. According to Segrè’s estimates, for bombarding-proton energies exceeding by 10–15% the threshold of the reaction \(p + p = 3p + \bar p\), the cross section for \(\bar p\) formation does not exceed \(1/400\) of the total cross section of nuclear interaction.

Next in line at present is the experimental discovery of the antineutron \(\bar n\). There is every reason to believe that \(\bar n\)’s are produced under the same conditions and in approximately the same quantity as \(\bar p\)’s. However, the technique of deflection by a magnetic field, which made it possible to determine the sign of the charge and the mass of the \(\bar p\), is not applicable to the \(\bar n\). The antineutron will have to be identified by its nuclear interactions. To the question “what is an antineutron” it is often answered that it is a neutral particle with a mass equal to the mass of the neutron, but with the opposite sign of the magnetic moment, just as the \(\bar p\) differs from \(p\) by the negative sign of its electric charge. Such a definition cannot be called erroneous, but it is very incomplete; it speaks of a certain particular property and does not mention the main, most essential one.

In reality the most important common property of the antinucleons—\(\bar n\) and \(\bar p\)—is their ability to annihilate with nucleons. Upon entering some nucleus, an antinucleon annihilates with one of the nucleons of the nucleus; in this process energy is released equivalent to twice the mass of the nucleon (plus the kinetic energy of motion of the antinucleon). This energy is spent on the kinetic energy of neutrons and protons flying out of the nucleus, and on the formation of mesons. At present there already exist the first examples of stars in photographic emulsion, caused by antiprotons, in which the energy of the star substantially exceeds the kinetic energy of the particle that caused the formation of the star[^11],[^14],[^44],[^45]. This property of antinucleons is most conveniently, clearly, and briefly formulated with the aid of the “nuclear charge”[^15].

II. NUCLEAR AND NEUTRINO CHARGE

AND THE LAW OF EQUIVALENCE OF MASS AND ENERGY

In all known radioactive processes the total number of nucleons is conserved. Taking into account the existence of antinucleons, this law should be supplemented so that what is conserved is the difference between the number of nucleons and the number of antinucleons. Assigning to each nucleon a nuclear charge \(+1\) (the same for the proton and the neutron) and to an antinucleon a nuclear charge \(-1\), one may speak of the conservation of total nuclear charge in any processes. Here there is a complete analogy with electric charge. Electric charge may be defined as the number of positively charged particles (charge \(+e\) for each) minus the number of negative particles (charge of the particle \(-e\)). Nuclear charge is assigned unambiguously to every elementary particle. Thus, for example, it is \(+1\) for the hyperon \(\Lambda^{0}\), which decays into \(p+\pi^{-}\), and is equal to 0 for \(\pi\)-mesons and \(\theta\)-mesons, which decay according to the reaction \(\theta=\pi^{+}+\pi^{-}\). The formulation “conservation of nuclear charge” includes the impossibility of such processes as the transformation of two or four neutrons into mesons—in whatever approximation, through any virtual stages. The fact of conservation of the number of nucleons is well known, but in connection with the discovery of hyperons and antinucleons it is useful to give it its most concise and convenient form.

Let us note that conservation of nuclear charge has a direct connection with the question of the equivalence of mass and energy, which was discussed in the pages of Uspekhi Fizicheskikh Nauk in 1952. Let us begin with an analogy. A system of 10 electrons has electric charge \(-10e\); by virtue of charge conservation, the lowest energy state of the system consists of 10 electrons at rest and situated at a large distance from one another, in order to reduce the Coulomb energy. In this state the system has the smallest mass, \(10m_e\). For an arbitrary initial state of the system, its mass \(M_0\) is greater than \(10m_e\). But only the excess energy \((M_0-10m_e)c^2\) can be taken away from the system and used externally; this excess is the kinetic and Coulomb energy of the electrons. Conservation of electric charge does not allow the energy corresponding to the rest mass of the electrons to be released and realized. If one takes a system of 9 electrons and 1 positron, with charge \(-8e\), the lowest state corresponds to 8 electrons with mass \(8m_e\), and a part of the energy corresponding to rest mass can be released*). Finally, in a neutral

*) The formation, upon annihilation of two particles \(e^{+}+e^{-}\), of two other particles—\(2\gamma\)-quanta, on which some philosophers often insist, is not fundamental, since \(\gamma\)-quanta can be absorbed without remainder, and it is not necessary. The energy released in annihilation can just as well be converted, for example, into heat, i.e. go into changing the kinetic energy of stable particles, and not into the birth of \(\gamma\)-quanta.

in a system of \(5e^+ + 5e^-\), all the energy can be released, including the energy corresponding to the entire rest mass.

If one takes an ordinary neutral substance, for example \(1\) g of hydrogen, then conservation of electric charge does not prevent its complete annihilation. However, conservation of nuclear charge means that such annihilation is impossible; only transformations conserving the number of nucleons are possible. In this case the lowest-energy state of matter with a number of nucleons equal to the number of nucleons in \(1\) g of hydrogen \((6.02 \cdot 10^{23}/1.00876)\) is \(\sim 0.985\) g of iron or elements close to it, as follows from measurements of mass defects. The release of energy in the transformation of \(1\) g of hydrogen into \(0.985\) g of iron corresponds to \(0.015\) g, i.e. amounts to about \(10^{19}\) ergs, which is approximately 15 times greater than the energy released in the fission of \(1\) g of uranium-235 in a nuclear reactor. Thus only about \(1.5\%\) of the energy corresponding to the rest mass can be released. Apparently, it was precisely this circumstance that V. A. Fock\(^{16}\) had in mind when he wrote: “But the overwhelming part of the energy (and the rest mass corresponding to it) usually does not participate in transformations and is conserved separately.” Only in a system consisting of electrons and positrons or mesons, or of an equal number of nucleons and antinucleons, i.e. in a system in which not only the electric but also the nuclear charge is zero, is complete release of the energy corresponding to the rest mass possible. In such a system the release of energy per \(1\) g would exceed by more than 1000 times the release of energy in the fission of \(1\) g of uranium-235. Any practical application of such systems is apparently hopeless, not only because their preparation would require an amount of electric power many times exceeding the annihilation energy. The point is also that it is practically difficult to imagine a practically acceptable method of accumulating and storing antinucleons in which they would not annihilate immediately, near the apparatus in which they are produced.

Returning to the main subject of the article, let us say that the basic distinction between a proton and an antiproton, a neutron and an antineutron lies in the sign of the nuclear charge, and not in the sign of the electric charge or magnetic moment. As an illustration one may cite another example: the hydrogen atom \(\mathrm H\), in the state with spin 0. Such an atom may be regarded as an elementary particle, electrically neutral and without a magnetic moment.

“Antihydrogen” \(\overline{\mathrm H}\), consisting of an antiproton and a positron, is likewise neutral and has no magnetic moment. But this does not mean the identity of \(\mathrm H\) and \(\overline{\mathrm H}\)! In vacuum each of these particles is stable, but upon contact with ordinary matter \(\mathrm H\) remains stable, whereas \(\overline{\mathrm H}\) annihilates immediately. This obvious difference between \(\mathrm H\) and \(\overline{\mathrm H}\) is most conveniently formulated by saying that \(\mathrm H\) has nuclear charge

\(+1\), while \(\bar{\mathrm H}\) has charge \(-1\). \(\mathrm H\) and \(\bar{\mathrm H}\) are not neutral with respect to nuclear charge, whereas “ordinary” matter has a positive nuclear charge.

The symmetry of the laws of nature with respect to charge conjugation implies a simultaneous change of sign of all charges. In this, \(e^{-}\) goes over into \(e^{+}\), \(p\) into \(\bar p\), \(n\) into \(\bar n\), \(\mathrm H\) into \(\bar{\mathrm H}\). Particles neutral with respect both to electric and to nuclear charge (\(\gamma\)-quanta and \(\pi^{0}\)-mesons) are thereby transformed into themselves. Let us note that one cannot change the sign of one of the charges without changing the signs of the other charges. This is already clear from the fact that there is no negatively charged nucleon (with positive nuclear charge).

A very instructive example is provided by charged hyperons: particles \(\Xi^{+}\) and \(\Xi^{-}\) are known, with different signs of electric charge, but both with positive nuclear charge. These two particles cannot annihilate one another; they are not a “particle and antiparticle” with respect to each other. It follows from the theory that there should also exist two particles with negative nuclear charge, \(\bar{\Xi}^{+}\) and \(\bar{\Xi}^{-}\), which have not been observed up to the present time. According to recent data (D. Steinberger, report at the conference in Moscow, May 1956), the masses of \(\Xi^{+}\) and \(\Xi^{-}\) are different, which also agrees with the fact that these two particles with opposite electric charge do not form a “particle–antiparticle” pair. In a “particle–antiparticle” pair all charges must have the opposite sign, not only the electric charge.

The twice-performed operation \(P\) of charge conjugation obviously returns the whole system to its initial state, so that \(P^{2}=+1\). At the same time, for particles which under the action of \(P\) are transformed into themselves, the operation \(P\) may multiply the wave function either by \(+1\) or by \(-1\); both possibilities are compatible with the general property \(P^{2}=+1\). Thus there arises the concept of charge-even neutral particles \(P=+1\) (example: \(\pi^{0}\)) and charge-odd particles \(P=-1\) (example: \(\gamma\)-quantum). For charged particles the concept of charge parity is inapplicable, since under a single application of \(P\) they are transformed into other particles. It can be shown that charge parity is a conserved quantity*).

Hence a whole series of strict selection rules is obtained: for example, a \(\pi^{0}\)-meson can decay only into an even number of \(\gamma\)-quanta. Charge parity can also be defined for a neutral system of particles. Thus, for example, an “atom” consisting of \(\pi^{+}\)- and \(\pi^{-}\)-mesons is, obviously, an even or odd system depending on the parity of the or-

*) Charge symmetry means the invariance of the Hamiltonian \(H\) with respect to \(P\), whence it follows that \(H\) and \(P\) commute, and hence \(P\) is an integral of motion. The exact symmetry with respect to charge conjugation considered here has no relation to the approximate, i.e., charge symmetry in nuclear physics of small energies, associated with the replacement of neutrons by protons and protons by neutrons (“mirror nuclei”).

of orbital angular momentum \(l\), since charge conjugation transforms \(\pi^+\) into \(\pi^-\), and \(\pi^-\) into \(\pi^+\), i.e. is equivalent to their spatial interchange. In an \(S\)-state such an atom is even; it can transform into 2 \(\gamma\)-quanta or into a \(2\pi^0\)-meson, or into \(\pi^0 + 2\gamma\)-quanta, but not into \(\pi^0 + 1\gamma\)-quantum. The combination of the exact principle of charge symmetry and the approximate theory of isotopic spin leads to a number of elegant and important non-strict selection rules; (see \(^{5}\) and the references cited in \(^{5}\)).

For the neutrino it is impossible to determine a priori whether charge conjugation leads to another particle—the antineutrino (as in the case of the electron and positron)—or whether the antineutrino coincides with the neutrino and there exists only one type of particle, as in the case of the \(\pi^0\)-meson or the \(\gamma\)-quantum. As was set out in detail in another article in UFN \(^{17}\), these two variants of the neutrino theory—the so-called Dirac theory and the Majorana theory—lead to essentially different predictions concerning the process of double \(\beta\)-decay. The work of McCarthy \(^{18}\), which appeared after the review \(^{17}\), seemed to speak in favor of the Majorana theory; however, according to Avshalom’s latest measurements \(^{19}\), the author failed to detect double \(\beta\)-decay

\[ \mathrm{Ca}^{48} \to \mathrm{Ti}^{48} + 2e^-, \]

and, by his estimate of the sensitivity of the apparatus, the decay period is greater than \(2 \cdot 10^{18}\) years. A large group of American researchers \(^{20}\) attempted to detect the double \(\beta\)-decay of neodymium 150

\[ (\mathrm{Nd}_{150}^{60} \to \mathrm{Sm}_{150}^{62} + 2e^- + 4.4\ \text{MeV}). \]

They also obtained a negative result (decay period greater than \(2 \div 4 \cdot 10^{18}\) years). From these works it follows that the neutrino and antineutrino are distinct particles. In this case, alongside electric and nuclear charge, one may introduce a third quantity, conserved in all transformations—the neutrino charge \(^{21}\).

By choosing in an appropriate way the neutrino charge of the electron and of the \(\mu\)-meson, one can strictly forbid processes such as the decay \(\mu^- = e^- + \gamma\) or \(\mu^- + n = n + e^-\), which are in fact not observed in carefully performed experiments. In a less precise formulation, analogous results were independently obtained by Konopinski and Mahmoud \(^{22}\).

In conclusion, let us note the special status of electric charge. The law of conservation of electric charge was discovered even in classical electrodynamics, without relation to electron theory; the electric charge of a body can be measured from the electrostatic field surrounding the body. The nuclear and neutrino charge of a body can be measured only by counting all elementary particles.

In ordinary matter, containing neither antinucleons nor free hyperons and mesons, an approximate measure of nuclear charge—the number of nucleons—is the mass of the body. Owing to the mass defect of nuclei this measure is not entirely exact, and in one gram of hydrogen there are fewer

nucleons as compared with 1 g of heavy elements. Let us note that the nuclear charge is not a quantity characterizing the interaction with mesons.

The existence of a neutrino charge, although it appears probable, requires further experimental confirmation. Let us note an interesting attempt by Marx to establish a connection between the neutrino charge and the constant characterizing the probability of processes in which neutrinos are produced[^23].

III. STRANGE PARTICLES

New ideas concerning antiparticles arose in 1954–1955 as a result of intensive experimental and theoretical investigations of new particles—hyperons and heavy mesons. From various sides these investigations have already been discussed in the pages of UFN[^24,^25,^26]. Therefore, in the present article we shall only briefly remind the reader of those basic facts which are necessary for what follows. All details, the analysis of the reliability of the experimental data, and the bibliography will be omitted, since they are available in the articles mentioned. We are interested only in information about the neutral hyperon \(\Lambda_0\) and the neutral meson \(\theta^0\). Since other particles are not considered here, the subscript “0” will be omitted.

These particles are characterized by the decay reactions

\[ \Lambda = p + \pi^- + 37\ \text{MeV}, \tag{1} \]

\[ \theta = \pi^+ + \pi^- + 214\ \text{MeV}. \tag{2} \]

Hence the mass of \(\Lambda = 2180\,m_e\), the mass of \(\theta = 963\,m_e\).
The decay period of \(\Lambda\): \(3.7\cdot 10^{10}\) sec, the period of \(\theta\): \(1.5\cdot 10^{10}\) sec.
The production of \(\Lambda\) and \(\theta\) occurs through the reactions

\[ \pi^- + p = \Lambda + \theta, \tag{3} \]

\[ N + N = \Lambda + \theta + N^*). \tag{4} \]

The production of one or two \(\Lambda\)-particles without \(\theta\) would require a considerably smaller expenditure of energy; the corresponding reactions would have a lower threshold. For what follows it is especially important that B. Pontecorvo and his co-workers, despite deliberate searches[^27], did not observe reactions (5) and (6).

\[ N + N \ne N + \Lambda, \tag{5} \]

\[ N + N \ne \Lambda + \Lambda. \tag{6} \]

The fact that \(\Lambda\)- and \(\theta\)-particles are produced only in pairs is naturally consistent with their long lifetime.

Ya. B. Zeldovich

To reactions of elementary particles one may apply a kind of algebra in which one can transfer a particle from one side of an equation to the other, replacing it by an antiparticle (see above, the relation between creation of a particle and annihilation of an antiparticle). The reaction equation may be read from left to right and from right to left in accordance with the principle of reversibility of all processes, or, as it is called in statistics and thermodynamics, the principle of detailed balance. In this case the probabilities of the direct and inverse reactions, referred to one level or to unit volume in phase space, are equal. In practice, if energies close to the reaction threshold are not considered specially, the cross sections of the direct and inverse processes are of the same order.

It is known from experiment that nucleons and \(\pi\)-mesons interact strongly with one another. Processes compatible with conservation of electric and nuclear charge, and involving only nucleons and \(\pi\)-mesons, occur with cross sections from \(10^{-24}\) to \(10^{-27}\ \text{cm}^2\). In those cases where the interaction of nucleons and \(\pi\)-mesons leads to the formation of bound states, the binding energy amounts to from several MeV to tens of MeV. Finally, such a strong interaction corresponds to decay periods from \(10^{-19}\ \text{sec}\) to \(10^{-22}\ \text{sec}^{*}\).

As an example let us point to an excited state of the nucleon: this state, to which spin \(3/2\) is assigned, is observed in the resonant form of the curve of meson scattering by nucleons \(^{8,12,28}\). The width of the resonance (about \(100\ \text{MeV}\)) corresponds to a lifetime of the order of \(10^{-23}\ \text{sec}\).

The strong interaction is characterized by the dimensionless number \(g^2/\hbar c\) of order unity; therefore repeated inclusion of the strong interaction corresponds to repeated multiplication by a quantity of order unity and does not substantially change the cross sections of processes and decay periods. It is easy to see that if reaction (5) for the single production of \(\Lambda\) proceeded with an appreciable cross section, one could construct from it the decay scheme

\[ \left. \begin{aligned} N+N &= N+\Lambda;\quad \Lambda = N+N+\overline{N};\\ \Lambda &= N+N+\overline{N} = N+\pi^{0} = p+\pi^{-} \end{aligned} \right\} \tag{7} \]

and one would have to expect a decay period of \(\Lambda\) of the order of \(10^{-20}\ \text{sec}\). With the aid of the same arguments, from the observed lifetime of \(\Lambda\), equal to \(3.7\cdot 10^{-10}\ \text{sec}\), one can estimate the cross section for single production of \(\Lambda\): this cross section must be \(10^{10}\)–\(10^{12}\) times smaller

\(*\) The estimate of periods is valid until special causes intervene that slow the decay of heavy compound nuclei, for example the Coulomb barrier for the emission of \(\alpha\)-particles or the large statistical weight of excited heavy nuclei, which slows the evaporation of neutrons.

cross sections of processes with strong interaction, i.e., is of the order of \(10^{-38}\,\text{cm}^2\) and is inaccessible to observation.

The facts of the slow decay of \(\Lambda\) and \(\theta\) are not independent; there is an internal connection between them owing to the fact that pair production of \(\Lambda\) and \(\theta\)—reactions (3) and (4)—corresponds to the strong interaction. One can construct, describing the decay of \(\theta\), a chain of “strong” virtual processes with one “weak,” slow link—the decay of \(\Lambda\).

\[ \theta = p+\pi^-+\bar{\Lambda} = p+\pi^-+\bar{p}+\pi^+ = \pi^-+\pi^+ . \tag{8} \]

In this chain the first (strong) stage is a consequence of reaction (3), the second (weak, slow) stage is the decay of \(\bar{\Lambda}\). By charge symmetry the decay of \(\bar{\Lambda}\) is characterized by the same weak interaction as the decay of \(\Lambda\). Finally, the last (strong) stage is the interaction of nucleons and \(\pi\)-mesons.

If one adopts the assumption that each virtual process with strong interaction contributes a factor of order 1, then from this follows the equality—in order of magnitude—of the decay times of \(\Lambda\) and \(\theta\).

With the same success one could, regarding the decay of \(\theta\) as the primary process, reduce the decay of \(\Lambda\) to it.

The views set forth make trivial the absence of single production of \(\Lambda\) (reaction (5)). However, the fact of the absence of pair production \(2\Lambda\) by reaction (6) is quite unexpected and leads to very important conclusions.

Indeed, \(\theta\) is a neutral particle with respect to all kinds of charges, like the \(\pi^0\)-meson or the \(\gamma\)-quantum. One might expect that under charge conjugation \(\theta\) transforms into itself, as does the \(\pi^0\)-meson or the \(\gamma\)-quantum. In that case \(\theta\) could be transferred from one side of the equation to the other, and one could construct a scheme for the production of \(2\Lambda\) from only strong interactions:

\[ N+N = N+\theta+\Lambda = p+\pi^-+\theta+\Lambda = 2\Lambda . \tag{9} \]

However, from experiment we know that such a process does not occur \(^{27}\). Consequently, the speculative assumption made above—that \(\theta\), under charge conjugation, transforms into itself—is incorrect. It is necessary to assume that there are two different neutral particles, \(\theta\) and \(\bar{\theta}\), i.e., that \(\theta\) has (distinct from \(\theta\) itself) an antiparticle \(\bar{\theta}\).

It is clear that the choice itself, which of them is to be called a particle and which an antiparticle, is conventional. We shall retain the name particle and the notation \(\theta\) for that particle which is produced in a pair with \(\Lambda\). For it the strong interaction is characteristic

\[ N = \theta+\Lambda . \tag{10} \]

It follows from this that there is a strong interaction

\[ \bar{\theta}+N=\Lambda. \tag{11} \]

However, there is no strong interaction

\[ \theta+N\ne \Lambda. \tag{12} \]

Thus, \(\theta\) is produced together with \(\Lambda\), but, on striking nuclei, cannot produce subsequent \(\Lambda\)'s.

In order to visualize the internal difference between \(\pi^\pm\), \(\pi^0\), \(\theta\), and \(\bar{\theta}\), let us consider the so-called polarization of the vacuum by these particles. According to quantum mechanics, as a result of interaction with other states, a particle must spend part of the time in these other states. Thus the \(\pi^+\)-meson spends part of the time in the form of a complex of a proton and an antineutron, \(p+\bar n\). As is known, Fermi and Yang even attempted to construct a theory according to which the \(\pi\)-meson is always such a pair\(^{29}\). This theory did not lead to quantitative results, but it does contain a vivid picture, and the rational kernel is that for part of the time the meson must be in the state of a nucleon–antinucleon pair. In this case the \(\pi^0\)-meson with equal probability gives the pairs \(p+\bar p\) and \(n+\bar n\). The nucleon–antinucleon pairs produced in the vacuum by \(\pi\)-mesons are in an \(S\)-state with spin 0. Under charge conjugation it is obvious that the pair \(p+\bar p\) transforms into itself, just as the pair \(n+\bar n\) does. Hence it is clear that under charge conjugation \(\pi^0\) transforms into itself.

M. A. Markov\(^{30,43}\) extends the Fermi–Yang theory to the \(\theta\)-meson; from equation (10) it follows that

\[ \theta=(N+\bar{\Lambda}). \]

Accordingly, the charge-conjugate particle is

\[ \bar{\theta}=(\bar N+\Lambda). \]

In principle, \(\theta\) can transform into \(\bar{\theta}\); however, such a process must occur through the slow stages of the decay of \(\Lambda\) into \(p+\pi^-\) and their charge-conjugate and inverse reactions (asterisks over the equality signs mark the slow stages):

\[ \theta=N+\bar{\Lambda}\overset{*}{=}N+\bar p+\pi^+=p+\bar N+\pi^-\overset{*}{=}\Lambda+\bar N=\bar{\theta}. \]

In Gell-Mann’s theory\(^{31,31a,32}\), which has successfully described the production and decay of many particles discovered in recent times, a new quantum number \(S\) (strangeness) is introduced. To each particle its own value of \(S\) is assigned, and it is postulated that processes proceeding with conservation of the sum \(S\) are characterized by strong interaction. Such processes

go in collisions; if a decay with conservation of \(S\) is possible, then such a decay takes place in a time \(\sim 10^{-20} — 10^{-23}\) sec, so that the corresponding particle cannot be observed in a free state.

Processes with a change of the sum \(S\) by one unit correspond to decay times of the order of \(10^{-10}\) sec; in collisions, lasting \(10^{-22}\) sec, such processes are never observed.

Finally, for processes with a change of the sum \(S\) by two units or more one would have to expect periods of the order of 1 sec, so that in practice such processes are not observed at all.

Nucleons and \(\pi\)-mesons are assigned \(S=0\), the hyperon \(\Lambda: S=+1\); hence for \(\theta: S=-1\) and for \(\bar{\theta}: S=+1\), in accordance with reactions (10) and (11). For what follows it is essential that the mutual transformation of \(\theta\) and \(\bar{\theta}\) cannot be forbidden by exact conservation laws (conservation of electric, nuclear, or neutron charge). According to the Gell-Mann theory this transformation is accompanied by a change of \(S\) by two units and consequently cannot proceed rapidly; it can occur only by virtue of a very weak interaction, but nevertheless this transformation is possible.

Digressing for a moment from the main topic, let us note that the experimental data \(^{33,34,46}\), apparently\(^*\), indicate a large spin of the \(\Lambda\)-particle; the plane of the decay \(\Lambda \to \pi^- + p\), as a rule, makes a small angle with the plane in which the \(\Lambda\)-particle was produced (in the center-of-mass system the trajectories of all four particles participating in the reaction \(\pi^+ + p = \Lambda + \theta\) lie in one plane).

The large spin of \(\Lambda\) must be taken into account in a concrete theory of the behavior and decay of \(\Lambda\); however, the hypotheses advanced earlier, which reduced the long lifetime of \(\Lambda\) exclusively to a large spin without assumptions about a weak interaction, are apparently incorrect. Such theories do not explain why there is no pair production \(2\Lambda\).

It is known that \(\Lambda\)-particles can enter into nuclei and decay there with an appreciable lifetime. If the slowness of the decay of a free \(\Lambda\)-particle were due to the large angular momentum which the \(\pi\)-meson formed in the decay must carry away, then in the nucleus the decay of \(\Lambda\) could proceed very rapidly by internal conversion \(^{35,36}\), with transfer of energy and angular momentum to another nucleon of the nucleus, without formation of a \(\pi\)-meson, which is not in agreement with experiment.

\(^*\) Note added in proof. D. Steinberger reported at the Conference on High-Energy Physics (Moscow, May 1956) that his measurements in a bubble chamber do not confirm the correlation of the planes of production and decay of \(\Lambda\). The reason for the discrepancy with the preceding works has not been clarified. Experimentally, the question of the spin of \(\Lambda\) remains open.

Let us note the curious remark of Dallaporta \(^{37,38}\), according to which the probability of decay of \(\Lambda\) and \(\theta\) is characterized by a coupling constant of the same order as the coupling constant of processes with neutrino emission, i.e., the \(\beta\)-process \(N=p+e^-+\nu\), capture of a \(\mu\)-meson, decay of a \(\mu\)-meson. Introducing a hypothetical interaction \(\Lambda=N+N+\overline{N}\), assigning to it the same constant as to the \(\beta\)-process, and considering the decay of \(\Lambda\) and \(\theta\) through this stage as a virtual state, Dallaporta obtains a satisfactory estimate of the lifetime of \(\Lambda\) and \(\theta\).

IV. FEATURES OF \(\theta\)-MESONS

The preceding arguments have shown that the neutral meson \(\theta\) has an antiparticle \(\bar{\theta}\), distinct from \(\theta\) itself. From this fact Gell-Mann and Pais \(^{39}\), and also Pais and Piccioni \(^{40}\), draw very interesting conclusions, which will be set forth below. The wave function \(\theta\) is not an eigenfunction of the charge-conjugation operator \(P\). The eigenfunctions of the charge-conjugation operator are the symmetric \((\theta_s)\) and antisymmetric \((\theta_a)\) linear combinations

\[ \theta_s=\frac{1}{\sqrt{2}}(\theta+\bar{\theta}), \]

\[ \theta_a=\frac{1}{\sqrt{2}}(\theta-\bar{\theta}). \]

Conversely, the wave function of the particle \(\theta\), from this point of view, should be regarded as a linear combination of \(\theta_s\) and \(\theta_a\)

\[ \theta=\frac{1}{\sqrt{2}}(\theta_s+\theta_a). \]

Thus, the production of a particle \(\theta\) in reality represents, with probability \(1/2\), the production of \(\theta_s\), and with probability \(1/2\), the production of \(\theta_a\).

In vacuum, in free flight, it is precisely \(\theta_s\) and \(\theta_a\), and not \(\theta\) and \(\bar{\theta}\), that should be regarded as independent distinct particles. Since the charge-conjugation operator commutes with the Hamiltonian, \(\theta_s\) and \(\theta_a\) have, generally speaking, different energies (on this in detail below). What is still more important, \(\theta_s\) and \(\theta_a\) must have different decay schemes.

Apparently,* the spin of \(\theta\) is 0; it is obvious that the spins of \(\bar{\theta}\), \(\theta_s\), and \(\theta_a\) coincide with the spin of \(\theta\). A pair of \(\pi^+\)- and \(\pi^-\)-mesons in an \(S\)-state

*) In the decay of \(\theta\), no correlation is observed \(^{33}\) between the direction of the decay products and the directions of the particles participating in the production of \(\theta\).

DEVELOPMENT OF THE THEORY OF ANTIPARTICLES AND THE PROPERTIES OF HEAVY MESONS

is a charge-even system. Consequently, only \(\theta_s\) can decay into \(\pi^+\) and \(\pi^-\), whereas \(\theta_a\) must seek other decay channels. Apparently, the lifetime of \(\theta_a\) is considerably longer than the lifetime of \(\theta_s\). Thus, Pais and Gell-Mann, who put forward this concept, arrive at the conclusion that half of the \(\theta\)-mesons produced together with \(\Lambda\) escape observation.

In a subsequent paper Pais and Piccioni consider in more detail the question of the nuclear interaction of \(\theta\)-mesons, taking into account considerations concerning \(\theta_s\) and \(\theta_a\). When \(\theta_s\) strikes a nucleus, it is necessary to regard \(\theta_s\) as a mixture of two kinds of particles: \(\bar{\theta}\), capable of giving a hyperon with a nucleon*) \((\bar{\theta}+N=\Lambda)\), and \(\theta\), capable only of being scattered, but not producing \(\Lambda\). The same also applies to \(\theta_a\), which, from the point of view of interaction with nuclei, is likewise a mixture of \(\theta\) and \(\bar{\theta}\).

In a beam containing both kinds of particles \(\theta_s\) and \(\theta_a\), in order to determine the numbers of \(\theta\) and \(\bar{\theta}\) it is necessary to take into account the phase relations (in quantum mechanics it is not probabilities that are added, but amplitudes). One must first add the wave functions of \(\theta_s\) and \(\theta_a\), and then determine the coefficients of \(\theta\) and \(\bar{\theta}\) in this sum. In considering decay it was sufficient to know that the particle produced together with the hyperon \(\theta\) is a mixture of two different particles \(\theta_s\) and \(\theta_a\); to obtain the complete picture it is necessary to remember the phase relation. The theory predicts a very peculiar picture of the decay and interaction with matter of \(\theta\)-mesons.

Let us imagine a thin target bombarded by \(\pi^-\)-mesons; in this target the reaction \(\pi^-+p=\Lambda+\theta\) takes place. The particles produced in the target do not give the reaction \(\theta+N=\Lambda\). Thus, in a second target placed next to the first (but so that the \(\pi^-\)-mesons do not enter it), the \(\theta\)-mesons will not cause the production of \(\Lambda\)-hyperons.

Let the \(\theta\)-mesons be given the opportunity to fly several centimeters in vacuum or in a gas, for example in a Wilson chamber. Along this path there will occur practically complete decay of the \(\theta_s\)-component \((\theta_s=\pi^++\pi^-)\), but the \(\theta_a\)-component will remain. If target III of dense substance is placed after the path over which the decay of \(\theta_s\) has ended, then in this target the mesons \(\theta_a\) will already be able to produce \(\Lambda\)-particles, since \(\theta_a\) contains \(\bar{\theta}\) (with probability \(1/2\): let us recall that the wave function \(\theta_a=\dfrac{1}{\sqrt{2}}(\theta-\bar{\theta})\)). In front of target III the beam consists of particles \(\theta_a\) alone; the decay of \(\theta_s\) into \(\pi^++\pi^-\) has ended earlier, while the decay of \(\theta_a\) into \(\pi^++\pi^-\) is impossible. Therefore, in front of

*) The excess energy may be absorbed by other nucleons of the nucleus or go into the formation of \(\pi\)-mesons.

target III, decays into \(\pi^{+}+\pi^{-}\) do not occur. In target III itself, as a result of the nuclear interaction, the component \(\bar{\theta}\) is absorbed from the \(\theta_a\) flux; consequently, at the exit from target III the beam contains an excess of \(\theta\) over \(\bar{\theta}\); such a beam no longer consists of \(\theta_a\) alone, but also contains a \(\theta_s\) component.* After target III, decays into \(\pi^{+}+\pi^{-}\) again appear, although in target III there had been only absorption, and not the production of new particles.

These peculiar relations can be illustrated by an analogy with the behavior of polarized light. Let \(\theta\) and \(\bar{\theta}\) correspond to the vertical and horizontal positions of the plane of polarization; \(\theta_s\) and \(\theta_a\) correspond to planes of polarization rotated by \(45^\circ\) (Fig. 1).** The decay corresponds to the absorption of the \(\theta_s\) component, i.e. to a polaroid rotated in such a way that only \(\theta_a\) passes through. In the initial beam of \(\theta\) there is no \(\bar{\theta}\) component, but after such a polaroid the \(\theta_a\) component already contains \(\bar{\theta}\). Interaction with nuclei in target III corresponds to a polaroid which absorbs \(\bar{\theta}\) and transmits \(\theta\), etc.

Fig. 1.

Fig. 2.

In addition to the decay of \(\theta_s\), it is also necessary to take into account the possible difference between the masses of \(\theta_s\) and \(\theta_a\); let us denote these masses by \(m_s\) and \(m_a\). The wave functions of \(\theta_s\) and \(\theta_a\) (in the system in which these particles are at rest) are proportional, respectively, to

\[ \exp ( i m_s c^2 t/\hbar ) \quad \text{and} \quad \exp ( i m_a c^2 t/\hbar ). \]

In the optical analogy, the mass difference corresponds to double refraction with axes \(\theta_s\) and \(\theta_a\), and rotates the plane of polarization of a beam which at the moment of production consisted of \(\theta\). Thus, if one measures the cross section for hyperon production in target III as a function of its distance \(l\) from the point of production of \(\theta\) (from the first target), one should expect an oscillatory course of the curve. In Fig. 2 the cross section of a pure \(\theta\) is taken as unity. The asymptotic value is \(1/2\).

* Not only absorption of \(\bar{\theta}\), but also scattering of \(\theta\) by nuclei as a result of a phase change, should lead to the formation of \(\theta_s\) in a beam which initially consisted only of \(\theta_a\).

** Pais and Piccioni use another system of notation and liken \(\theta_s\) and \(\theta_a\) to circularly polarized light.

corresponds to a beam of \(\theta_a\) after complete decay of \(\theta_s\). The period of the oscillations in flight time is equal to \(2\pi\hbar/c^2(m_s-m_a)\), where \(m_s\) and \(m_a\) are the masses of \(\theta_s\) and \(\theta_a\).

Thus, in principle it proves possible to measure the negligibly small expected mass difference between \(\theta_s\) and \(\theta_a\).

Recently, \(41\) cases have been observed of the anomalous decay of \(\theta\), apparently into a \(\mu\)-meson, a \(\pi\)-meson, and a neutrino. It should be expected that such a decay is possible both for \(\theta_s\) and for \(\theta_a\); however, for \(\theta_s\) it will be little noticeable because of competition from the principal type of decay, \(\theta_s=\pi^++\pi^-\), while for \(\theta_a\) the decay into \(\mu\pi\nu\) will be the principal process. Obviously, each of the components \(\theta_s\) and \(\theta_a\) separately gives, in equal numbers, \(\mu^+\pi^-\nu\) and \(\mu^-\pi^+\nu\). In the initial period, when both \(\theta_s\) and \(\theta_a\) are present, one should expect \(^{42}\) an oscillatory behavior of the ratio

\[ \frac{\mu}{\mu_+ + \mu_-} \]

similar to the curve in Fig. 2. How can one estimate the order of magnitude of the possible mass difference? Pais and Piccioni, without proof, give the estimate \(\Delta m\sim \hbar/\tau c^2\), where \(\tau\) is the lifetime of \(\theta_s\). We present arguments \(^{42}\) confirming this estimate.

The mass difference of \(\theta_s\) and \(\theta_a\) depends on the possibility of the transformation \(\theta\rightleftarrows\bar{\theta}\). Such a transformation, accompanied by a change of strangeness by 2 units, is a process of higher order, weaker in comparison with the decay \(\theta\to\pi^++\pi^-\). At first sight it follows from this that the mass difference should be considerably smaller than the decay probability multiplied by \(\hbar/c^2\). In reality, the decay probability is proportional to the square of the matrix element for a process with a change of strangeness \(\Delta S=1\), i.e. proportional to \(g^2\), where \(g\) is the coupling constant. Meanwhile the mass difference is proportional to the first power of the matrix element for the transition \(\theta\rightleftarrows\bar{\theta}\) with a change of strangeness \(\Delta S=2\). Indeed, if one writes symbolically

\[ -i\frac{\partial \theta}{\partial t}=E_0\theta+f\bar{\theta};\qquad -i\frac{\partial \bar{\theta}}{\partial t}=E_0\bar{\theta}+f\theta, \]

we obtain

\[ E_s=E_0+f, \]
\[ E_a=E_0-f; \]

since we are dealing with a perturbation of a degenerate system, \(E_0\) for \(\theta\) and \(\bar{\theta}\) are identically equal. One may expect that \(f\sim g^2\), so that \(\Delta m\sim \dfrac{\hbar}{\tau c^2}\), where \(\tau\) is the decay period \(\sim 1.5\cdot 10^{-10}\) s in agreement with \(^{40}\). Numerically we obtain

\[ \Delta m=10^{-11}m_e, \]

where \(m_e\) is the electron mass.

Another approach to the question of the difference of the masses \(\theta_s\) and \(\theta_a\) is based on a direct consideration of that coupling of the \(\theta\)-particles with other fields which causes the decay. If it is assumed that the spin of \(\theta\) is zero, then the pair \(\pi^+,\pi^-\) formed in the decay is in a state even with respect to charge conjugation, and only the decay \(\theta_s=\pi^+ + \pi^-\) is possible, but not the decay \(\theta_a\). The decay \(\theta_s\) testifies to the coupling of the field \(\theta_s\) with the field of \(\pi\)-mesons (coupling constant \(g\)). According to the usual formulas of perturbation theory, such a coupling must cause a shift of the level, i.e. a change in the energy of \(\theta_s\), along with the decay, which causes a broadening of the level. The change in the energy and mass of \(\theta_s\) cannot be directly expressed through the decay time, but from formulas \(^{42}\) it is seen that the coupling constant \(g\) enters into \(\Delta m\) and into \(1/\tau\) in the same degree, which again confirms the estimate of Pais and Piccioni.

In expounding the theory of \(\theta_s\) and \(\theta_a\), following Pais and Gell-Mann \(^{39}\), we began with the formal properties of the charge-conjugation operator \(P\). For a theoretical physicist such a theory is absolutely clear. More “makeshift,” but perhaps more visual and convincing, the same results can be obtained by making only the single assumption that \(\bar{\theta}\), just like \(\theta\), is capable of decaying into \(\pi^+\) and \(\pi^-\). In reality, of course, this assumption contains the principle of charge symmetry. Let \(\Pi(E)\) denote the wave function of the pair \(\pi^+ + \pi^-\) with energy \(E\). It is obvious that \(\Pi(E)\) belongs to the continuous spectrum and that the energies of \(\theta\) and \(\bar{\theta}\) are identical; denote them by \(E_0=m_\theta c^2\). The wave equations, without taking the decay into account, have the form

\[ -i\hbar \frac{\partial \theta}{\partial t}=E_0\theta \]

and analogously for \(\bar{\theta}\). Taking into account the decay of \(\theta,\bar{\theta}\) into \(\pi^+ + \pi^-\), it is necessary to introduce into the consideration the field of \(\pi\)-meson pairs and its coupling with the field \(\theta\). The equations have the form

\[ -i\hbar \frac{\partial \theta}{\partial t} = E_0\theta+\int M(E)\Pi(E)\rho(E)\,dE, \]

\[ -i\hbar \frac{\partial \bar{\theta}}{\partial t} = E_0\bar{\theta}+\int M(E)\Pi(E)\rho(E)\,dE, \]

\[ -i\hbar \frac{\partial \Pi(E)}{\partial t} = E\cdot \Pi(E)+M(E)\theta+M(E)\bar{\theta}. \]

Here \(M(E)\) is the matrix element for the transformation of \(\theta\) into a pair \(\pi^+ + \pi^-\) with energy \(E\). All states are considered, the whole spectrum of pairs \(\pi^+ + \pi^-\); \(\rho(E)\) is the weight function giving the density of states per unit energy interval.

The equations are symmetric with respect to \(\theta\) and \(\bar{\theta}\). If they are solved under asymmetric initial conditions at \(t=0\), \(\theta=1\), \(\bar{\theta}=0\), \(\Pi(E)=0\), then the mathematics itself leads to all the conclusions described—

DEVELOPMENT OF THE THEORY OF ANTIPARTICLES AND THE PROPERTIES OF HEAVY MESONS

... above; in particular, it follows from the equations that the square of the amplitude \(\bar{\theta}\), equal to 0 at the beginning, will increase, oscillating along the curve of Fig. 2, etc.

All the experimental conclusions can be obtained without saying that the particles are \(\theta_s\) and \(\theta_a\); however, the equations also show the convenience of introducing the sum and difference of \(\theta\) and \(\bar{\theta}\), which simplifies the equations; in particular, it is seen that for the difference \((\theta_a)\) an equation without decay holds.

CITED LITERATURE

1. I. Ya. Pomeranchuk, DAN 60, 213 (1948).
2. L. D. Landau, DAN 60, 207 (1948).
3. E. M. Lifshitz, DAN 60, 211 (1948).
4. A. Ore and J. L. Powell, Phys. Rev. 75, 1696 (1949).
5. I. S. Shapiro, UFN 53, 7 (1954).
6. A. Hedgran and D. A. Lind, Phys. Rev. 82, 126 (1951).
7. L. A. Page, P. Stehle and S. B. Gunst, Phys. Rev. 89, 1273 (1953).
8. H. Bethe, F. de Hoffman, Mesons and Fields. II. 1955.
9. K. M. Crowe, R. H. Phillips, Phys. Rev. 96, 470 (1955).
10. S. Koslov, V. Fitch, J. Rainwater, Phys. Rev. 95, 291 (1954).
11. O. Chamberlain, E. Segrè, C. Wiegand, T. Ypsilantis, UFN 58, 685 (1956).
12. H. A. Bethe, UFN 56, 149 (1955).
13. S. Z. Belenkii and L. D. Landau, UFN 56, 309 (1955).
14. E. Segrè et al., Nuovo Cimento 3, 447 (1956).
15. Ya. B. Zel’dovich, DAN 86, 505 (1952).
16. V. A. Fock, UFN 48, 161 (1952).
17. Ya. B. Zel’dovich, S. Yu. Luk’yanov, Ya. A. Smorodinskii, UFN 54, 361 (1954).
18. J. A. McCarthy, Phys. Rev. 97, 1234 (1955).
19. Awschalom, Bull. Am. Phys. Soc. No. 1 (1956).
20. C. L. Cowan, F. B. Harrison, L. M. Langer, F. Reines, A Test of Neutrino—Antineutrino Identity, Nuovo Cim. 3, 649 (1956).
21. Ya. B. Zel’dovich, DAN 91, 1317 (1953).
22. E. J. Konopinski, H. M. Mahmoud, Phys. Rev. 92, 1045 (1953).
23. G. Marx, Zs. Naturforschung 9a, 1051 (1954).
24. A. I. Alikhanov, UFN 50, 481 (1953).
25. S. F. Powell, UFN 53, 449 (1954).
26. A. O. Weisenberg, UFN 57, 361 and 631 (1955).
27. M. P. Balandin, B. D. Balashov, V. A. Zhukov, B. M. Pontecorvo, G. I. Selivanov, ZhETF 29, 265 (1955).
28. I. E. Tamm, Yu. A. Gol’fand, V. Ya. Fainberg, UFN 26, 649 (1954).
29. E. Fermi, C. Yang, Phys. Rev. 76, 1739 (1949).
30. M. A. Markov, On the Systematics of Elementary Particles. Publishing House of the Academy of Sciences, Moscow, 1955.
31. M. Gell-Mann, Phys. Rev. 92, 833 (1953).
    31a. M. Gell-Mann, A. Pais, Proc. Glasgow Conference, 1954.
32. M. Gell-Mann, Proc. Pisa Conference, 1955.

1. W. B. Fowler, R. P. Shutt, A. M. Thorndike, W. L. Whittemore, Phys. Rev. 93, 861 (1954); R. P. Shutt, Proceedings V Rochester Conference 100 (1955).
2. Walker and Shepherd—cited by R. Gatto, Nuovo Cimento 3, 499 (1956).
3. R. Gatto, Nuovo Cimento 1, 372 (1955).
4. I. Yu. Kobzarev, L. B. Okun, ZhETF 30, 798 (1956).
5. N. Dallaporta, Nuovo Cimento 1, 962 (1955).
6. G. Costa and N. Dallaporta, Nuovo Cimento 2, 519 (1955).
7. M. Gell-Mann and A. Pais, Phys. Rev. 97, 1387 (1955).
8. A. Pais and O. Piccioni, Phys. Rev. 100, 1487 (1955).
9. R. W. Thompson, Progress in Cosmic Rays III (cited from 38).
10. Ya. B. Zel'dovich, ZhETF 31, June (1956).
11. M. A. Markov, DAN 106, 814 (1956).
12. R. D. Hill et al., Phys. Rev. 101, 907 (1956).
13. O. Chamberlain et al., Phys. Rev. 101, 909 (1956).
14. J. Ballam et al., Phys. Rev. 97, 245 (1955).
15. W. H. Barkas, W. Birnbaum, M. Smith, Phys. Rev. 101, 778 (1956).

1. \(N\) is a neutron. ↩