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Pauli and the Pauli group
Nuclear Physics B (Proc. Suppl.) 137 (2004) 1–4 www.elsevierphysics.com
Pauli and the Pauli group K. Nishijima Nishina Memorial Foundation 8-28-45 Hon-Komagome, Bunkyo-ku, Tokyo 113-8941, Japan Reminiscences of Pauli and of the PauIi group or his last contribution to neutrino physics are briefly described.
1. THE CONCEPT GROUP
OF
THE
PAULl
Neutrino was invented by Pauli (1900—1958) in order to save conservation of energy and consequently also of momentum and angular momentum and he remained the godfather of neutrino for life. In 1956 he was shocked by learning that parity is not conserved in weak interactions and that it is maximally violated whenever a neutrino participates. This was probably one of the motivations for his introduction of the Pauli group since it clarifies why violation of parity had not beeen recognized before in beta- decay. 1.1. Philosophy behind the Pauli group In connection with beta-dacay Pauli introduced a new concept which we shall refer to as the Pauli group hereafter. Before its explicit introduction we shall present a generalized version of his philosophy behind the Pauli group. Let L(gi ; ψj ) be the Lagrangian density of a system of fields, where ψj denotes a field operator and gi a fundamental parameter such as a mass or a coupling constant,. Let us introduce a group G of canonical transformations of the field operators ψj → ψj ,
(1)
such that the change of L caused by (1) can be compensated by the corresponding transformations of the parameters gi → gi .
(2)
O(gi ) = O(gi ).
(3)
This implies that O(gi ) must be a function of g s invariant under G. The group G has to be chosen in accordance with the experimental conditions.
Namely, we have L(gi ; ψj ) = L(gi ; ψj ).
Then there are two possible cases: 1) When the transformation of gi belongs to the one-dimentional representation of the group G, namely, gi = gi , we have an invariance and the corresponding conservation law based on Noether’s theorem. 2) Otherwise, when (2) is not trivial and gi = gi , we have no invariance under the group G. In what follows we shall discuss the consequences of (3) in this case. Given a set of parameters gi and a set of field operators ψj , we are going to evaluate some observable quantities under certain experimental conditions. The process of evaluating them is in some sense similar to that of cooking a dish. The set of parameters are ingredients of the dish such as meat, vegetables, etc., and the set of field operators are kitchen utensils such as pots and pans since they are carried away in the last stage of presenting the dish or the observable quantity. Indeed, in preparing a dish we can choose any set of pots and pans as long as we follow the given recipe faithfully. The final product, a dish or an observable quantity, contains only the ingredients initially prepared or the set of parameters in the Lagrangian. Thus, we can evaluate an observable quantity by starting from either side of (3) so that an observable quantity O must satisfy the condition
0920-5632/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2004.10.046
(4)
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K. Nishijima / Nuclear Physics B (Proc. Suppl.) 137 (2004) 1–4
In the following subsections we shall give two such examples.
The Lagrangian is also invariant under a U(1) group GII ,
1.2. A simple example As an example we introduce a free Lagrangian density of a lepton field ψ by
ψ → ψ = eiαγ5 ψ,
¯ µ ∂µ + (m1 + iγ5 m2 )]ψ, Lf (mi ; ψ) = −ψ[γ
(5)
and then assume that the interaction Lagrangian is invariant under the following U(1) group ψ = eiα(1−γ5 ) ψ,
ψ¯ = ψ¯ eiα(1+γ5 )
(6)
In this case the total Lagrangian L satisfies the equality L(m1 , m2 ; ψ ) = L(m1 , m2 ; ψ)
(7)
m1 = m1 cos 2α − m2 sin 2α,
(9)
1.3. The Pauli group The second example is the one presented by Pauli [1] for the massless neutrino with the free Lagrangian, ¯ µ ∂µ ψ. Lf = −ψγ
(10)
This Lagrangian is invariant under an SU(2) group GI . ¯ ψ → ψ = aψ + bγ5 C ψ, ψ¯ → ψ¯ = a∗ ψ¯ − b∗ C −1 γ5 ψ,
(15)
In what follows we shall use a new notation ψ ψ , = Ψ= −γ5 C ψ¯ γ5 γ4 Cψ † ¯ −ψC −1 γ5 ), Ψ¯ = (ψ, (16)
( )
and (14) as Ψ → Ψ = eiαγ Ψ.
( )
There are three generators for SU(2) and one for U(1), and they are given by 1 d3 x(ψ † γ5 γ4 Cψ † + ψC −1 γ4 γ5 ψ), M1 = 2 1 d3 x(ψ † γ5 γ4 Cψ † − ψC −1 γ4 γ5 ψ), M2 = 2i M3 = d3 xψ † ψ, N = d3 xψ † γ5 ψ. (17)
They satisfy the following commutation relations: (11)
[
with |a|2 + |b|2 = 1.
G = GI × GII .
(8)
Then all the observable quantities must be functions of the invariant combination of m1 and m2 ,namely, m2 = m21 + m22 .
(14)
These two sets of transformations commute with each other and the entire group G is given by their direct product, namely,
then (11) can be expressed as a −b Ψ → Ψ = Ψ, b∗ a∗
provided that m2 = m1 sin 2α + m2 cos 2α,
ψ¯ → ψ¯ = ψ¯ eiαγ5
Mk Mi M j , [Mi , N ] = 0. ] = i ijk , 2 2 2
(18)
We are using the Pauli metric here except for the charge conjugation matrix that is actually the inverse of Pauli’s, namely,
They generate the group G and satisfy the relations: 0 1 Ψ, [Ψ, Mi ] = σi Ψ, e.g. [Ψ, M ] = 1 0
C −1 γµ C = −γµT , C T = −C.
[Ψ, N ] = γ Ψ,
(12)
(13)
(19)
K. Nishijima / Nuclear Physics B (Proc. Suppl.) 137 (2004) 1–4
Now in terms of Ψ we have 1 Lf = − Ψ¯ γµ ∂µ Ψ, 2 1 d3 xΨ † σi Ψ, Mi = 2 1 d3 xΨ † γ Ψ, N= 2 Ψ¯ Ψ = .
(20)
The beta-decay interaction may be expressed as Lβ = (ψ¯p , Oj ψn )(ψ¯e , Oj (fj† + γ5 gj† )Ψ ) j
+ h. c.
(21)
where j runs over five types of Dirac matrices and fj1 gj1 fj = , gj = , fj2 gj2 ∗ ∗ , fj2 ), fj† = (fj1
∗ ∗ gj† = (gj1 , gj2 ),
(22)
We have compared evaluation of observable quantities to cooking and assumed that there is no trace of the employed pots and pans in the final product or the dish. In some cases , however, some gourmet might recognize what kinds of pots and pans were used while tasting the dish by smelling the delicate flavor. This is indeed the case when there is a conserved quantity, and we could change pots and pans provided that the quantum number is respected. That means reduction of the group G to its subgroup G1 that leaves this quantum number unchanged. As an example, we shall choose N − M3 as the conserved quantity, then only N and M3 commute with N − M3 so that G1 is given by G1 = U (1) × U(1)
fj1 = gj1 , fj2 = −gj2
L(f , g ; Ψ ) = L(f, g; Ψ ).
fj1 = ei(λ+α) fj1 , fj2 = e−i(λ+α) fj2 .
Under the experimental condition that neutrinos are not observed and the helicity of the electrons not measured, any observable quantity must be a function of f and g through their combinations invariant under G , namely, fi † fj + gi† gj , fi† gj − gi† fj , fi1 gj2 − fj2 gi1 (24) Take, for instance, the vector coupling (Oj → γµ ) we find G2F = fV† fV + gV† gV . The old choice of fV and gV is given by 0 1 fV = GF , , gV = 0 0 whereas the new choice is given by 1 −1/2 fV = gV = 2 GF . 0
(25)
(26)
(26 )
The distinction between them became possible only after measuring the helicity of electrons or of neutrinos.
(27)
These U(1) groups are generated by M3 and N , respectively. Furthermore, this conservation law implies the relation
Then the total Lagrangian remains unchanged when Ψ and f + γ5 g transform in the same way under G = SU(2) × U(1). (23)
3
(28)
The parameters fj are transformed under G1 as (29)
Their combinations invariant under G1 are given by ∗ ∗ ∗ ∗ fj1 , fi2 fj2 , fi1 fj2 , fi1 fj2 fi1
(30)
In this case the neutrino is left-handed and the antineutrino right-handed. Measurements of the helicity of emitted neutrinos [2] indicate that they are left-handed suggesting the absence of antineutrinos, namely, fj2 = 0.
(31)
Measurements of the electron helicity [3,4] as combined with the above result led us to the V-A theory. Finally it should be mentioned that the Lagrangian density (20) in terms of Ψ is suitable for introduction of the non-abelian gauge field corresponding to the Pauli group. 2. ENCOUNTER WITH PAULI In 1956 I was at the Max-Planck-Institute for Physics then in Goettingen working on the
4
K. Nishijima / Nuclear Physics B (Proc. Suppl.) 137 (2004) 1–4
subject mentioned in Subsection 1.2. One day Heisenberg warned me that Pauli had some objections to that work and that I should be careful if I were to face him. A disaster came sooner than anticipated. In April 1957 I visited the Niels Bohr Institute in Copenhagen and gave a seminar on the subject in issue. It was then and there that I first met Pauli. Since the content of my work was not about symmetry itself it must have sounded rather ambiguous, and I suddenly recalled Heisenberg’s warning when I was confronted with Pauli’s attack on my talk. Then Pauli gave a talk entitled ”On the Conservation of the Leptonic Charge” [1] which was briefly described in the Subsection 1.3. During his seminar Niels Bohr asked a question to Pauli who responded fiercely :”Do you still believe that energy is not conserved in beta-dacay?” That was all that I could recall on that day. Few months later I attended a meeting in Oberwohlfach in Schwarzwald. The meeting was held in a small castle which accomodated about forty participants. It was a meeting on field theory and related subjects organized by the Goettingen group, namely, Lehmann, Symanzik, Zimmermann and Haag. It was attended by celebrities such as Heisenberg, Pauli, Heitler, Kallen, Jost, Jensen, Mottelson and others. It so happened that I had to give the same talk again in front of Pauli. He kept silence this time, however, and after my talk he told me that it was my turn to attack him during his talk. I do not know whether he thought that my presentation had been improved or that a verdict of guilty should not be duplicated. At that time he had been interested in a certain aspect of Heisenberg’s non-linear spinor theory but their collaboration had been left in embryo. In the summer of 1957 I moved from Goettingen to Princeton to work at the Institute for Advanced Study. In January 1958 I attended the New York Meeting of the American Physical Society. A special session was held at Columbia University and Pauli gave a lecture on Heisenberg’s theory. Pauli was once interested in this theory and tried to collaborate with Heisenberg but he finally changed his mind. That was the last time
to see him for me. In December 1958 I attended a party at the Institute. Chen Ning Yang was there to pose a question: ”What is the mass of the quantum of the Yang-Mills field?” Nobody knew the answer to this question nor its significance at that time, but this question was related to Pauli’s comment on the Yang-Mills field. It was also in that party that I learnt of Pauli’s death during an operation. The year 1958 came to a close [5]. To conclude I would like to thank Prof. H. Miyazawa for his interest in this work and for his help in preparing this manuscript. REFERENCES 1. W.Pauli, Nuovo Cimento Serie X 6(1957)204. 2. M.Goldhaber, L.Grozins and A.W.Sunyar, Phys. Rev.109 (1958)1015. 3. H. Frauenfelder et al, Phys. Rev. 106 (1957) 386. 4. L. A. Page and M. Heinberg, Phys. Rev. 106 (1957)1220. 5. K. v. Meyenn and E. Schucking, Physics Today, Feb. 2001, 43.
Pauli and the Pauli group K. Nishijima Nishina Memorial Foundation 8-28-45 Hon-Komagome, Bunkyo-ku, Tokyo 113-8941, Japan Reminiscences of Pauli and of the PauIi group or his last contribution to neutrino physics are briefly described.
1. THE CONCEPT GROUP
OF
THE
PAULl
Neutrino was invented by Pauli (1900—1958) in order to save conservation of energy and consequently also of momentum and angular momentum and he remained the godfather of neutrino for life. In 1956 he was shocked by learning that parity is not conserved in weak interactions and that it is maximally violated whenever a neutrino participates. This was probably one of the motivations for his introduction of the Pauli group since it clarifies why violation of parity had not beeen recognized before in beta- decay. 1.1. Philosophy behind the Pauli group In connection with beta-dacay Pauli introduced a new concept which we shall refer to as the Pauli group hereafter. Before its explicit introduction we shall present a generalized version of his philosophy behind the Pauli group. Let L(gi ; ψj ) be the Lagrangian density of a system of fields, where ψj denotes a field operator and gi a fundamental parameter such as a mass or a coupling constant,. Let us introduce a group G of canonical transformations of the field operators ψj → ψj ,
(1)
such that the change of L caused by (1) can be compensated by the corresponding transformations of the parameters gi → gi .
(2)
O(gi ) = O(gi ).
(3)
This implies that O(gi ) must be a function of g s invariant under G. The group G has to be chosen in accordance with the experimental conditions.
Namely, we have L(gi ; ψj ) = L(gi ; ψj ).
Then there are two possible cases: 1) When the transformation of gi belongs to the one-dimentional representation of the group G, namely, gi = gi , we have an invariance and the corresponding conservation law based on Noether’s theorem. 2) Otherwise, when (2) is not trivial and gi = gi , we have no invariance under the group G. In what follows we shall discuss the consequences of (3) in this case. Given a set of parameters gi and a set of field operators ψj , we are going to evaluate some observable quantities under certain experimental conditions. The process of evaluating them is in some sense similar to that of cooking a dish. The set of parameters are ingredients of the dish such as meat, vegetables, etc., and the set of field operators are kitchen utensils such as pots and pans since they are carried away in the last stage of presenting the dish or the observable quantity. Indeed, in preparing a dish we can choose any set of pots and pans as long as we follow the given recipe faithfully. The final product, a dish or an observable quantity, contains only the ingredients initially prepared or the set of parameters in the Lagrangian. Thus, we can evaluate an observable quantity by starting from either side of (3) so that an observable quantity O must satisfy the condition
0920-5632/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2004.10.046
(4)
2
K. Nishijima / Nuclear Physics B (Proc. Suppl.) 137 (2004) 1–4
In the following subsections we shall give two such examples.
The Lagrangian is also invariant under a U(1) group GII ,
1.2. A simple example As an example we introduce a free Lagrangian density of a lepton field ψ by
ψ → ψ = eiαγ5 ψ,
¯ µ ∂µ + (m1 + iγ5 m2 )]ψ, Lf (mi ; ψ) = −ψ[γ
(5)
and then assume that the interaction Lagrangian is invariant under the following U(1) group ψ = eiα(1−γ5 ) ψ,
ψ¯ = ψ¯ eiα(1+γ5 )
(6)
In this case the total Lagrangian L satisfies the equality L(m1 , m2 ; ψ ) = L(m1 , m2 ; ψ)
(7)
m1 = m1 cos 2α − m2 sin 2α,
(9)
1.3. The Pauli group The second example is the one presented by Pauli [1] for the massless neutrino with the free Lagrangian, ¯ µ ∂µ ψ. Lf = −ψγ
(10)
This Lagrangian is invariant under an SU(2) group GI . ¯ ψ → ψ = aψ + bγ5 C ψ, ψ¯ → ψ¯ = a∗ ψ¯ − b∗ C −1 γ5 ψ,
(15)
In what follows we shall use a new notation ψ ψ , = Ψ= −γ5 C ψ¯ γ5 γ4 Cψ † ¯ −ψC −1 γ5 ), Ψ¯ = (ψ, (16)
( )
and (14) as Ψ → Ψ = eiαγ Ψ.
( )
There are three generators for SU(2) and one for U(1), and they are given by 1 d3 x(ψ † γ5 γ4 Cψ † + ψC −1 γ4 γ5 ψ), M1 = 2 1 d3 x(ψ † γ5 γ4 Cψ † − ψC −1 γ4 γ5 ψ), M2 = 2i M3 = d3 xψ † ψ, N = d3 xψ † γ5 ψ. (17)
They satisfy the following commutation relations: (11)
[
with |a|2 + |b|2 = 1.
G = GI × GII .
(8)
Then all the observable quantities must be functions of the invariant combination of m1 and m2 ,namely, m2 = m21 + m22 .
(14)
These two sets of transformations commute with each other and the entire group G is given by their direct product, namely,
then (11) can be expressed as a −b Ψ → Ψ = Ψ, b∗ a∗
provided that m2 = m1 sin 2α + m2 cos 2α,
ψ¯ → ψ¯ = ψ¯ eiαγ5
Mk Mi M j , [Mi , N ] = 0. ] = i ijk , 2 2 2
(18)
We are using the Pauli metric here except for the charge conjugation matrix that is actually the inverse of Pauli’s, namely,
They generate the group G and satisfy the relations: 0 1 Ψ, [Ψ, Mi ] = σi Ψ, e.g. [Ψ, M ] = 1 0
C −1 γµ C = −γµT , C T = −C.
[Ψ, N ] = γ Ψ,
(12)
(13)
(19)
K. Nishijima / Nuclear Physics B (Proc. Suppl.) 137 (2004) 1–4
Now in terms of Ψ we have 1 Lf = − Ψ¯ γµ ∂µ Ψ, 2 1 d3 xΨ † σi Ψ, Mi = 2 1 d3 xΨ † γ Ψ, N= 2 Ψ¯ Ψ = .
(20)
The beta-decay interaction may be expressed as Lβ = (ψ¯p , Oj ψn )(ψ¯e , Oj (fj† + γ5 gj† )Ψ ) j
+ h. c.
(21)
where j runs over five types of Dirac matrices and fj1 gj1 fj = , gj = , fj2 gj2 ∗ ∗ , fj2 ), fj† = (fj1
∗ ∗ gj† = (gj1 , gj2 ),
(22)
We have compared evaluation of observable quantities to cooking and assumed that there is no trace of the employed pots and pans in the final product or the dish. In some cases , however, some gourmet might recognize what kinds of pots and pans were used while tasting the dish by smelling the delicate flavor. This is indeed the case when there is a conserved quantity, and we could change pots and pans provided that the quantum number is respected. That means reduction of the group G to its subgroup G1 that leaves this quantum number unchanged. As an example, we shall choose N − M3 as the conserved quantity, then only N and M3 commute with N − M3 so that G1 is given by G1 = U (1) × U(1)
fj1 = gj1 , fj2 = −gj2
L(f , g ; Ψ ) = L(f, g; Ψ ).
fj1 = ei(λ+α) fj1 , fj2 = e−i(λ+α) fj2 .
Under the experimental condition that neutrinos are not observed and the helicity of the electrons not measured, any observable quantity must be a function of f and g through their combinations invariant under G , namely, fi † fj + gi† gj , fi† gj − gi† fj , fi1 gj2 − fj2 gi1 (24) Take, for instance, the vector coupling (Oj → γµ ) we find G2F = fV† fV + gV† gV . The old choice of fV and gV is given by 0 1 fV = GF , , gV = 0 0 whereas the new choice is given by 1 −1/2 fV = gV = 2 GF . 0
(25)
(26)
(26 )
The distinction between them became possible only after measuring the helicity of electrons or of neutrinos.
(27)
These U(1) groups are generated by M3 and N , respectively. Furthermore, this conservation law implies the relation
Then the total Lagrangian remains unchanged when Ψ and f + γ5 g transform in the same way under G = SU(2) × U(1). (23)
3
(28)
The parameters fj are transformed under G1 as (29)
Their combinations invariant under G1 are given by ∗ ∗ ∗ ∗ fj1 , fi2 fj2 , fi1 fj2 , fi1 fj2 fi1
(30)
In this case the neutrino is left-handed and the antineutrino right-handed. Measurements of the helicity of emitted neutrinos [2] indicate that they are left-handed suggesting the absence of antineutrinos, namely, fj2 = 0.
(31)
Measurements of the electron helicity [3,4] as combined with the above result led us to the V-A theory. Finally it should be mentioned that the Lagrangian density (20) in terms of Ψ is suitable for introduction of the non-abelian gauge field corresponding to the Pauli group. 2. ENCOUNTER WITH PAULI In 1956 I was at the Max-Planck-Institute for Physics then in Goettingen working on the
4
K. Nishijima / Nuclear Physics B (Proc. Suppl.) 137 (2004) 1–4
subject mentioned in Subsection 1.2. One day Heisenberg warned me that Pauli had some objections to that work and that I should be careful if I were to face him. A disaster came sooner than anticipated. In April 1957 I visited the Niels Bohr Institute in Copenhagen and gave a seminar on the subject in issue. It was then and there that I first met Pauli. Since the content of my work was not about symmetry itself it must have sounded rather ambiguous, and I suddenly recalled Heisenberg’s warning when I was confronted with Pauli’s attack on my talk. Then Pauli gave a talk entitled ”On the Conservation of the Leptonic Charge” [1] which was briefly described in the Subsection 1.3. During his seminar Niels Bohr asked a question to Pauli who responded fiercely :”Do you still believe that energy is not conserved in beta-dacay?” That was all that I could recall on that day. Few months later I attended a meeting in Oberwohlfach in Schwarzwald. The meeting was held in a small castle which accomodated about forty participants. It was a meeting on field theory and related subjects organized by the Goettingen group, namely, Lehmann, Symanzik, Zimmermann and Haag. It was attended by celebrities such as Heisenberg, Pauli, Heitler, Kallen, Jost, Jensen, Mottelson and others. It so happened that I had to give the same talk again in front of Pauli. He kept silence this time, however, and after my talk he told me that it was my turn to attack him during his talk. I do not know whether he thought that my presentation had been improved or that a verdict of guilty should not be duplicated. At that time he had been interested in a certain aspect of Heisenberg’s non-linear spinor theory but their collaboration had been left in embryo. In the summer of 1957 I moved from Goettingen to Princeton to work at the Institute for Advanced Study. In January 1958 I attended the New York Meeting of the American Physical Society. A special session was held at Columbia University and Pauli gave a lecture on Heisenberg’s theory. Pauli was once interested in this theory and tried to collaborate with Heisenberg but he finally changed his mind. That was the last time
to see him for me. In December 1958 I attended a party at the Institute. Chen Ning Yang was there to pose a question: ”What is the mass of the quantum of the Yang-Mills field?” Nobody knew the answer to this question nor its significance at that time, but this question was related to Pauli’s comment on the Yang-Mills field. It was also in that party that I learnt of Pauli’s death during an operation. The year 1958 came to a close [5]. To conclude I would like to thank Prof. H. Miyazawa for his interest in this work and for his help in preparing this manuscript. REFERENCES 1. W.Pauli, Nuovo Cimento Serie X 6(1957)204. 2. M.Goldhaber, L.Grozins and A.W.Sunyar, Phys. Rev.109 (1958)1015. 3. H. Frauenfelder et al, Phys. Rev. 106 (1957) 386. 4. L. A. Page and M. Heinberg, Phys. Rev. 106 (1957)1220. 5. K. v. Meyenn and E. Schucking, Physics Today, Feb. 2001, 43.