## Über die Erhaltungssätze in der Quantenmechanik [On the conservation laws of quantum mechanics]. Offprint from: Nachrichten der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, 1927.

Berlin: Weidmann, 1928.

First edition, very rare offprint, of the invention of spatial parity as a quantum mechanical conserved quantity. “Wigner was invited to Göttingen in 1927 to become Hilbert's assistant. Hilbert, already interested in quantum mechanics, felt that he needed a physicist as an assistant to complement his own expertise. This was an important time for Wigner who produced papers of great depth and significance, introducing in his paper ‘On the conservation laws of quantum mechanics’ (1927) the new concept of parity” (mathshistory.standrews.ac.uk/Biographies/Wigner.html). “The concept of parity, which is very important for the understanding of spectra, has no analogy in classical theory comparable to the analogy between the orbital quantum number and the angular momentum” (Wigner, *Group Theory and its Application to the Quantum Mechanics of Atomic Spectra* (2012), p. 182). “Wigner was a member of the race of giants that reformulated the laws of nature after the quantum mechanics revolution of 1924-25. In a series of papers on atomic and molecular structure, written between 1926 and 1928, Wigner laid the foundations for both the application of group theory to quantum mechanics and for the role of symmetry in quantum mechanics” (David J. Gross, ‘Symmetry in Physics: Wigner’s legacy,’ *Physics Today*, December 1995, pp. 46-50). Wigner was awarded the Nobel Prize in Physics in 1963 “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles.” “Among Wigner’s accomplishments was the recognition that symmetry principles explained patterns found in atomic and molecular spectra. Wigner’s analysis of the application of mathematical group theory enabled physicists to understand relative stability or instability of nuclear isotopes having the same number of protons in the nucleus but different neutron numbers. Nearly a decade after he was awarded the Nobel Prize, Wigner’s early group theory research was described as so farsighted that it was not immediately recognized for its importance as a pioneering advance in mathematical physics. Interestingly referenced in the Nobel Prize presentation to Wigner was his law of the conservation of parity developed at Göttingen in 1928. The parity law states that particles emitted during a physical process should emanate from the left and right in equal numbers or equivalently that a nuclear process should be indistinguishable from its mirror image. The parity concept was not challenged until 1956 when it was disproved in certain so-called ‘weak decay’ interactions in experiments by Tsung-Dao Lee of Columbia and Chen Ning Yang of Princeton. Lee and Yang were awarded the Nobel Prize in 1957 for their empirical refutation of Wigner’s parity theory in this special case. The theory however remained substantially intact and along with other of Wigner’s discoveries useful as a further guide in nuclear research” (DSB). “It is scarcely possible to overemphasize the role played by symmetry principles in quantum mechanics” (C. N. Yang, Nobel Lecture, p. 394). No copies on OCLC or ABPC/RBH.

*Provenance*: Felix Bloch (1905-83), Swiss-American physicist who shared the 1952 Nobel Prize for Physics with Edward Purcell for “their development of new ways and methods for nuclear magnetic precision measurements” (‘Bloch’ written in ink on front wrapper). In 1928, the year the present paper was published, Bloch was awarded his doctorate under Werner Heisenberg for a brilliant thesis which established the quantum theory of solids, using ‘Bloch waves’ to describe electrons in periodic lattices.

The concept of parity refers to the behavior of classical and quantum systems under the ‘inversion’ operation, which takes a point in three dimensions with Cartesian coordinates *x*, *y*, *z* to the point with coordinates –*x*, –*y*, –*z* (more generally, this can be any ‘linear transformation’ that is not a rotation, for example the ‘mirror reflection’ that takes *x*, *y*, *z* to –*x*, *y*, *z*). Symmetry under inversion, or reflection, was used in classical physics, but was not of any great practical importance there. One reason for this derives from the fact that right-left symmetry is a discrete symmetry, unlike rotational symmetry which is continuous. In a famous paper in 1918, Emmy Noether showed that continuous symmetries always lead to conservation laws in classical physics – but a discrete symmetry does not. With the introduction of quantum mechanics, however, this difference between discrete and continuous symmetries disappears.

Wigner was led to his study of parity by work of Otto Laporte in 1924. Laporte studied the structure of the spectrum of iron and found that there are two kinds of energy levels, which he called ‘stroked’ (‘gestrichene’) and ‘unstroked’ (‘ungestrichene’). He discovered a selection rule (later called Laporte’s rule) that the transitions occurred always from stroked to unstroked levels or vice versa, and never between stroked or between unstroked levels. A few months later similar observations on the spectrum of titanium were made by Henry Norris Russell. No convincing explanation of the existence of two types of levels was found within the framework of the old quantum theory. In 1927, Wigner analyzed Laporte’s finding and showed that the two types of levels and the selection rule followed from the invariance of the electromagnetic forces in the atom under the operation of inversion of coordinates. This led him quickly to the idea of parity conservation in quantum mechanics. He wrote, ‘But that was very easy. I knew the spectroscopic rules, and Laporte’s rule was similar to the theory of inversion’ (quoted in *Collected Works*, Vol. 7, p. 7).

Wigner introduced the parity operator, and parity conservation, formally in the present paper, “a programmatic essay, entitled ‘*Über die Erhaltungssätze in der Quantenmechanik*’ (‘On the conservation laws of quantum mechanics’), which Max Born presented to the Göttingen Academy on 10 February 1928. In this paper, Wigner described the relation between the unitary operators *O*, which commute with the energy operator [Hamiltonian] *H*, and the conservation laws for physical quantities. The operators *O*, having the property that *Oψ* [where *ψ* is the wave-function of the quantum system] is an eigenfunction with the same eigenvalue *ε* as *ψ*, or

*H*(*Oψ*) = *εψ* if *Hψ* = *εψ*,

allow in special cases the following visualizable (*anschaulische*) interpretation:

- The space-translation operator is connected with the centre-of-mass theorem (or momentum conservation).
- The orthogonal rotation operation is connected with angular-momentum conservation.

Only the space-reflection operator (connected with the concept of parity), which can also be introduced here, does not seem to play a role in classical physics and therefore lacks proper visualization. In general, however, the theory of group representations can be closely tied to the known conservation laws, and they are ‘obtained from unitary transformations leaving the energy operator invariant’ ([the offered paper], p. 381)” (Mehra & Rechenberg, *Historical Development of Quantum Theory*, vol. 6 (2000), p. 509).

“Having pointed out that the commutation of displacement and rotation operators with the Hamiltonian of a system leads to the conservation of the position of the centre of mass and the conservation of the total angular momentum of the system, Wigner turned to the inversion operation. He stated the corresponding conservation law in the following way: The probability for a state to be labeled *g* [see below] or *u* (*Sp*_{+} and *Sp*_{–} in Wigner’s notation) does not change with time provided the Hamiltonian commutes with the parity operation. To illustrate its significance, Wigner considered bringing an H nucleus (a proton) towards an H atom, ultimately forming He^{+} … The initial state of the system, with respect to its centre of mass, has a probability of ½ of being of type *u* [and ½ of being of type *g*]; this distribution of probabilities must persist as the two nuclei are brought together, with the result that the final state of He^{+} must have an equal probability of being type *S* (the lowest *g* state) or type *P* (the lowest *u *state). As Wigner pointed out, this example has no classical analogue” (*Collected Works*, Vol. 1, p. 26).

In modern terminology, the ‘parity operator’ *Π* in quantum mechanics takes a wave-function *ψ *to the wave-function* ψ*(–*x*, –*y*, –*z*). Since *Π*^{2} takes every wave-function *ψ *to itself, it follows that if *ψ* is an ‘eigenfunction’ of *Π*, which means that *Πψ = λψ *for some number* λ *(the ‘eigenvalue’), we must have *λ*^{2 }= 1, and so *λ* = ±1. If *λ* = 1, the state *ψ *is said to have ‘positive parity’ and is labeled *g*; if *λ* = –1, *ψ* is said to have ‘negative parity’ and is labeled *u*. The law of conservation of parity for a quantum system with Hamiltonian *H* is the statement that the operators *H* and *Π *commute:* HΠ = ΠH. *

Wigner’s explanation of Laporte’s rule on the basis of parity conservation runs as follows. “The states that Laporte called stroked or unstroked are in fact states of positive parity and negative parity. Because the intrinsic parity of the emitted photon [resulting from the atomic transition that gives rise to the spectral line in question] is negative, then in order for the total parity of the system to be conserved, the parity of the atomic state must change. Thus, we get radiation emitted only when the atom changes from a state of positive parity to one of negative parity, or vice versa. Transition between states of the same parity are forbidden by parity conservation. This principle of parity conservation rapidly became an established principle of physics. As Frauenfelder and Henley state, ‘Since invariance under space reflection is intuitively so appealing (why should a left- and right-handed system be different?), conservation of parity quickly became a sacred cow.’

“That was the situation in the physics community until 1956” (Franklin, *The Neglect of Experiment* (1989), p. 11), when Lee and Yang proposed that parity is not conserved in weak interactions – although it conserved in electromagnetic and strong interactions. “In attempting to understand some puzzles in the decay of subatomic particles called K-mesons, the Chinese-born physicists Tsung-Dao Lee and Chen Ning Yang proposed in 1956 that parity is not always conserved. For subatomic particles three fundamental interactions are important: the electromagnetic, strong, and weak forces. Lee and Yang showed that there was no evidence that parity conservation applies to the weak force. The fundamental laws governing the weak force should not be indifferent to mirror reflection, and, therefore, particle interactions that occur by means of the weak force should show some measure of built-in right- or left-handedness that might be experimentally detectable. In 1957 a team led by the Chinese-born physicist Chien-Shiung Wu announced conclusive experimental proof that the electrons ejected along with antineutrinos from certain unstable cobalt nuclei in the process of beta decay, a weak interaction, are predominantly left-handed—that is to say, the spin rotation of the electrons is that of a left-handed screw” (Britannica).

The results of Wu’s experiment came as a shock to many physicists. “Even as late as 17 January 1957, Pauli had not given up his belief [in parity conservation]. In a letter to Viktor Weisskopf, he wrote that ‘I do not believe that the Lord is a weak left-hander, and I am ready to bet a very large sum that the experiments will give symmetric results.’ In a letter to C. S. Wu dated 19 January 1957, after hearing word of the results of her experiment, he noted that ‘I did not believe in it [parity nonconservation] when I read the paper of Lee and Yang.’ Frauenfelder and Henley were previously quoted as stating that parity conservation quickly became a sacred cow. They added later that the suggestion [that parity was not conserved] was ‘rejected by most physicists.’ They also reported an interesting anecdote: ‘R. P. Feynman, for instance, bet N. F. Ramsay $50 to $1 that parity is conserved. Feynman paid.’ Professor Lee also reported that Felix Bloch, another Nobel Prize winner [who inscribed our offprint of Wigner’s paper], offered to bet other members of the Stanford Physics Department his hat that parity was conserved. He later remarked to Lee that it was fortunate that he didn’t own a hat” (Franklin, p. 24).

It is now understood that processes involving weak interactions in which neutrinos are produced are likely to violate parity conservation. This is because neutrinos always spin ‘backward’ (in other words, the axis of their spin points away from their direction of motion), while antineutrinos spin ‘forward’ (their axis of spin points straight ahead). That means there are very subtle differences in the numbers of neutrinos and antineutrinos produced in a regular, versus a mirror-image, experiment that involves the weak nuclear force.

Born in Budapest, Wigner (1902-95) began his university career in chemistry at the insistence of his father, who wanted him to work in his tannery business. After one year at the Technical Institute in Budapest he transferred to the Technische Hochschule in Berlin where, alongside his chemistry work, Wigner studied physics and mathematics on his own. He attended the physics colloquium and witnessed first-hand the emerging understanding of quantum mechanics. During his third year in Berlin, Wigner began working at the Kaiser Wilhelm Institute in the suburb of Dahlem, where he met Michael Polanyi, a physical chemist who was also a native of Budapest. Polanyi agreed to be Wigner’s thesis advisor for a doctoral dissertation in chemical engineering that contained the first theory of rates of disassociation and association of molecules. After receiving his doctoral degree in 1925, the twenty-two-year-old Wigner dutifully returned home to Budapest to help his father at the tannery. After just one year, however, he accepted an assistantship set up by Polanyi with the x-ray crystallographer Karl Weissenberg at the University of Berlin. Recognizing Wigner’s fine command of mathematics, Weissenberg assigned him a problem that required an exploration of the elementary aspects of group theory. After a few months, Weissenberg arranged for Wigner to work with Richard Becker, who had recently been given a chair at the university in theoretical physics. In 1927 Becker, in turn, suggested that Wigner work with David Hilbert at the University of Göttingen. But Hilbert became ill and retreated from professional work, leaving Wigner without formal responsibilities. Wigner’s time in Göttingen was hardly unproductive, however … at the suggestion of fellow Hungarian Leo Szilard, Wigner began a book, *Group Theory and its Application to Quantum Mechanics *(1931), which became famous. By the time he left Göttingen, Wigner had firmly launched a career in science. Not only had he begun the book that would make his name, he had started the line of research that would later lead to his award of the Nobel Prize.

“Wigner contributed in a seminal way to theoretical physics. He is distinguished from other physicists who laid the foundations of quantum mechanics from the mid-1920s by his pioneering application of group theory and symmetry principles to quantum mechanics… Steven Weinberg has seen in the introduction of symmetry principles to physics the core of the revolutionary world view which has come to dominate modern physics: ‘Out of the fusion of relativity with quantum mechanics has evolved a new view of the world, one in which matter has lost its central role. This role has been usurped by the principles of symmetry, some of them hidden from view in the present state of the universe …’ Wigner was a key figure in building the new language Weinberg is writing about, especially in the domains of quantum mechanics and fundamental particle theory. While Wigner was not alone in carrying out this momentous project, he was a pioneer in applying symmetry considerations and group theory on a grand scale to quantum mechanics… while Hermann Weyl is sometimes thought of as the originator of the group-theoretic approach in quantum mechanics, Wigner’s groundbreaking papers on group theory appeared before Weyl’s important work on the subject” (M. Chayut, ‘From the Periphery: the genesis if Eugene P. Wigner’s application of group theory to quantum mechanics,’ *Foundations of Chemistry* 3 (2001), 55-78).

8vo (236 x 169 mm), pp. [375], 376-381, [1]. Original printed wrappers (sunned, a little creased at fore-edge).

Item #5005

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Price:
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