Simple derivation of general Fierz-type identities
C. C. Nishi
Citation: American Journal of Physics 73, 1160 (2005); doi: 10.1119/1.2074087 View online: http://dx.doi.org/10.1119/1.2074087
View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/73/12?ver=pdfcov Published by the American Association of Physics Teachers
Simple derivation of general Fierz-type identities
C. C. Nishia兲Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, São Paulo, SP, Brazil
共Received 17 February 2005; accepted 26 August 2005兲
General Fierz-type identities are examined and their well-known connection with completeness relations in matrix vector spaces is shown. In particular, I derive the chiral Fierz identities in a simple and systematic way by using a chiral basis for the complex 4⫻4 matrices. Other completeness relations for the fundamental representations of SU共N兲 algebras can be extracted using the same reasoning. ©2005 American Association of Physics Teachers.
关DOI: 10.1119/1.2074087兴
I. INTRODUCTION
The Fierz identities1are frequently used in particle physics to analyze four-fermion operators2 such as current-current operators. These reordering identities are used to write a product of two Dirac bilinears3 as a linear combination of other products of bilinears with the four constituent Dirac spinors in a different order. Because Fierz identities relate Dirac bilinears, which are objects with well defined transfor-mation properties under the Lorentz group, and Dirac gamma matrices form a representation of the Lorentz group genera-tors, it is not surprising that Fierz identities imply the exis-tence of a basis of four-vectors formed by Dirac bilinears and guarantee its orthogonality and completeness properties.4 The converse is also possible, that is, recovering the spinor from that basis, which reveals the equivalence between spinor and tensorial representation of various quantities.
Fierz identities are only a particular set of matrix identi-ties, valid for the Dirac bilinears which span the space of 4⫻4 complex matrices. General Fierz-type identities can be found for any N⫻N real or complex square matrices. The primary aim of this article is to show that any Fierz-type relation can be deduced using just a few ingredients such as the notion of a basis in a vector space and its properties of orthogonality and completeness. The relevant vector space in this case is the vector space of square matrices. For such a vector space, a basis and the orthogonality property between its elements can be defined through an inner product. These ideas are explained in detail in Sec. II for general square matrix vector spaces. Some particularly useful examples in-volving Pauli matrices关SU共2兲 algebra兴, Gell-Mann matrices 关SU共3兲 algebra兴, and fundamental representations of general SU共N兲 algebras are given.
As a particular nontrivial application of these ideas, we will find a straightforward way to deduce the chiral Fierz identities, that is, Fierz identities involving bilinears contain-ing left/right chiral projectors. One way to obtain the chiral Fierz identities is to use the original Fierz identities; one can find very general Fierz identities for usual bilinear in Ref. 4. However, this procedure may become quite lengthy because it requires expanding the left/right projected bilinears in terms of the usual bilinears, performing Fierz reorderings, and then rewriting them as projected bilinears. Such an ap-proach was adopted in Ref. 5, but will not be pursued here. Instead, the chiral Fierz identities will be deduced by red-eriving the Fierz identities using appropriate left/right pro-jected bilinears as a basis 共Sec. IV兲. The simple form of certain chiral Fierz identities is already an indication that a much simpler procedure should exist to derive them.
To exemplify the practical importance of chiral Fierz iden-tities, I will show a situation where the reordering permitted by chiral Fierz identities can help us to better understand and more simply describe a physical system. The obvious area where the chiral Fierz identities can be important is the phys-ics involving the weak interaction, which is a mainly left-handed interaction. The particular physical system concerns the neutrinos propagating through ordinary matter. Because ordinary matter contains electrons but is absent of muons and tauons, only the electron neutrinos interact with the electrons in matter through the charged-current effective interaction Lagrangian6
4GF
冑
2 ¯e共x兲␥␣Le共x兲¯e共x兲␥␣Le共x兲, 共1兲
wheree,eare the fields for the electron and the electron
neutrino, respectively, and L =共1−␥5兲/2 is the projector for left chirality. Using a chiral Fierz identity we can rewrite Eq. 共1兲 in the form
4GF
冑
2 ¯ e共x兲␥ ␣L e共x兲 ¯ e共x兲␥␣Le共x兲. 共2兲This reordered form enables us to describe the influence of nonrelativistic electrons in matter by their average density 2具¯
e共x兲␥␣Le共x兲典⬃␦␣0ne. Such a term leads to an effective
interaction acting on electron neutrinos that is different from the interactions acting on other types of neutrinos. Ulti-mately, it leads to a significant modification of the descrip-tion of neutrino oscilladescrip-tions,6 the phenomenon responsible for the missing solar neutrino problem.7
Before we get into the details, let us consider the differ-ence between the expressions in Eqs. 共1兲 and 共2兲 to under-stand better the nonintuitive nature of the Fierz identity. The spinor fields e and e can be written as complex 4⫻1
matrices, while¯eand¯eare complex 1⫻4 matrices. The
factors between them are 4⫻4 matrices and the result is a scalar. In Eq.共1兲, there is a 4⫻4 matrix between¯eande
and another between¯
eandefor each␣= 0 , 1 , 2 , 3. What
the Fierz identities assure you is that any expression of the form¯1A2¯3B4is equal to¯1C4¯3D2or a sum of simi-lar terms, with A , B⫽C,D in general. Amazingly, the same combination of matrices that enters into Eq. 共1兲 also enters into Eq.共2兲 as a consequence of a Fierz identity.
II. MATRIX VECTOR SPACES AND COMPLETENESS RELATIONS
A vector space V is a set of elements endowed with two operations:8A sum between elements and a multiplication by numbers共elements of a ring, in mathematical language兲 such as the real and complex numbers. This vector space is re-quired to be closed under such operations.9 The usual N-dimensional real space,RN, and its complex extension,CN,
are examples of vector spaces.
As is well known, any element of a vector space can be expanded in terms of a basis兵ei其, a set of N=dim V elements.
In addition, the vector space can be equipped with an inner product共 , 兲 that defines the notion of norm and orthogonal-ity. By using such an inner product, an orthonormal basis兵ei其
can be defined by
共ei,ej兲 =␦ij. 共3兲
The canonical 共column vector兲 representation for the ortho-normal basis is
共ei兲j=␦ij, 共4兲
where the index j outside the parenthesis is the vector index. In matrix notation the orthogonality relation共3兲 can be writ-ten as
ei
T
ej=␦ij, 共5兲
where T denotes the transpose. In canonical form, the com-pleteness relation
兺
i N eiei T = 1, 共6兲is obvious. Also, Eq. 共6兲 is invariant under an orthogonal O共N兲 关unitary U共N兲兴 transformation of a basis for V =RN共CN兲.
Once the properties of vector spaces are given, it is easy to show that the set of all N⫻N square matrices over the reals, MN共R兲, or over the complex numbers, MN共C兲, form a N2
dimensional vector space.10 In these vector spaces a canoni-cal basis兵eij其 is given by the matrices
共eij兲 kl=␦k
i␦ l
j 共i, j,k,l = 1, ... ,N兲, 共7兲
and hence, any N⫻N matrix can be expanded as
M = Mijeij, 共8兲
where the expansion coefficients Mij=共M兲ijare the elements
of the matrix M.
In MN共R兲 we can define the 共positive definite兲 inner
prod-uct
共A,B兲 ⬅ Tr关ABT兴, 共9兲
for which the canonical basis satisfies
Tr关eijeklT兴 =␦ik␦jl. 共10兲
For MN共C兲 the transpose operation 共 兲Thas to be replaced by
the hermitian conjugation operation共 兲†.
An equivalent approach is to define a bilinear function on MN共R兲 without the positive definiteness requirement of an
inner product. Thus, instead of defining Eq.共9兲, we can dis-card the transpose operation in the definition and regard sim-ply the trace of the product as the relevant bilinear function.
The missing transpose operation can be transferred to the definition of a dual basis 兵eij其: eij⬅eji= eijT. Then the dual
basis is the orthogonal counterpart of the basis兵eij其 through
the trace bilinear. An equivalent way is to regard the trace bilinear between two elements of 兵eij其 as defining a metric
that can be used to lower and raise indices and to interchange the basis with its dual; an analogous construction is found in special relativity when contravariant and covariant four-vectors are defined. An inner product defines, with an appro-priate basis, a metric that is the identity matrix. In general there can be nondiagonal or nonpositive definite metrics. This approach will be used to derive the Fierz identities in Sec. III and chiral Fierz identities in Sec. IV.
We use this dual basis to express the expansion coefficient in Eq.共8兲 as
Mij= Tr关Meij兴. 共11兲
If we substitute Eq.共11兲 into Eq. 共8兲 and take the respective elements of the matrix, we obtain the trivial relation
␦km␦nl=共eij兲kl共eij兲nm, 共12兲
where the summation convention of repeated indices is used here and in the following. This relation follows directly from Eq. 共4兲 and is a completeness relation analogous to Eq. 共6兲. Equation 共12兲 represents an identity in the space of general linear transformations over MN共R兲. This entire discussion is
also valid for MN共C兲 if one extends the ring from R to C.
Because any linear transformation over MN共R兲 can be given
as a linear combination of transformations of the form M→ 共A丢B兲M ⬅ AMBT=共A兲ij共B兲lk共M兲jke
il
, 共13兲 Eq. 共12兲 implies that
共eij丢eij兲M = 共eij丢eij兲M = M. 共14兲
Although trivial with this choice of basis, a completeness relation like Eq. 共12兲 is all that is needed to deduce Fierz-type identities.
To obtain nontrivial relations, let us take the example of SU共2兲 and SU共3兲 algebras in the fundamental representation. The commonly used representations for these algebras are the Pauli matrices兵i其 and the Gell-Mann matrices 兵a其.11,12
They form vector spaces and satisfy the orthogonality rela-tions
Tr关ij兴 = 2␦ij 共i, j = 1,2,3兲, 共15兲
Tr关ab兴 = 2␦ab 共a,b = 1, ... ,8兲. 共16兲
Because they are already orthogonal and are Hermitian ma-trices, there is no need to define a dual basis. However, to span M2共C兲 and M3共C兲 the respective identity matrices are needed, because N2 basis vectors are required and the Pauli and Gell-Mann matrices are traceless. Then the set 兵1,i其
spans M2共C兲, which means any 2⫻2 complex matrix can be expanded in terms of X = X01 + Xi i , i=i, 共17兲 where X0= 1 2Tr关X兴, Xi= 1 2Tr关Xi兴. 共18兲
We substitute Eq. 共18兲 into Eq. 共17兲 and take the general elements
1161 Am. J. Phys., Vol. 73, No. 12, December 2005 C. C. Nishi 1161
共X兲ij=
1
2共X兲kk␦ij+
1
2共X兲lk共m兲kl共m兲ij, 共19兲
and obtain from the coefficients of 共X兲lk, after properly
in-serting Kronecker deltas, the completeness relation
␦il␦kj=
1 2␦ij␦kl+
1
2共m兲ij共m兲kl. 共20兲
Equation 共20兲 is the identity used to deduce the Fierz iden-tities to Weyl spinors: 共兲ij共˜兲kl= 2␦il␦kj, where
=共1,兲, and˜=共1,−兲; the lowering and raising of in-dices follows the Minkowski metric.
For the Gell-Mann matrices we have similarly, 1
2共a兲ij共a兲kl+
1
3␦ij␦kl=␦il␦kj. 共21兲
For any fundamental representation of SU共N兲 algebra 兵Ta其
satisfying Tr关TaTb兴=C␦ab, we have the completeness relation
1
C共Ta兲ij共Ta兲kl+ 1
N␦ij␦kl=␦il␦kj. 共22兲 For the O共N兲 groups there is no simple relation similar to Eq.共22兲 because the algebra is formed by N⫻N antisymmet-ric matantisymmet-rices, and thus all symmetantisymmet-ric matantisymmet-rices are needed to span MN共R兲.
Before introducing Dirac matrices to deduce the Fierz identities, it is better to introduce a clearer notation due to Takahashi,4 where we replace the matrix indices by paren-theses共 兲 and brackets 关 兴, such that each parenthesis/bracket represents a different index in an unambiguous way. For ex-ample, using this notation the relation共22兲 reads
1
C共Ta兲关Ta兴 + 1
N共 兲关 兴 = 共 兴关 兲, 共23兲 where the blank entry means the identity matrix. This nota-tion clearly shows the reordering property.
III. FIERZ IDENTITIES
The starting point to derive the usual Fierz identities is the orthogonality relation among the 16 Dirac bilinears13 that span M4共C兲 over C. The 16 Dirac bilinears are usually clas-sified into distinct classes according to their properties under Lorentz transformations3as
兵⌫A其 = 兵1,␥
5,␥,␥5␥,其 共,= 0,1,2,3兲, 共24兲 where⬍ in to avoid redundancy. Here the conven-tion used by Itzykson and Zuber15 is employed for the gamma matrices: 兵␥,␥其 = 2g, 共25兲 = i 2关␥ ,␥兴, 共26兲 ␥5=␥ 5= i␥0␥1␥2␥3, 共27兲 ␥5= i 2 ␣ ␣, 共28兲
where 0123= 1 = −0123 and g= diag共1,−1,−1,−1兲 is the Minkowski metric.
Then, defining a basis兵⌫A其 dual to Eq. 共24兲 as the
respec-tive gamma matrices with space-time indices lowered by Minkowski metric,14 the orthogonality relation holds:
Tr关⌫A⌫B兴 = 4␦A B
. 共29兲
This relation allow us to expand any complex 4⫻4 matrix X in terms of the basis共24兲 as
X = XA⌫A, XA=
1
4Tr关X⌫A兴. 共30兲
We combine Eqs. 共29兲 and 共30兲, extract each element of the matrix, and find a completeness relation analogous to Eq. 共23兲: 共 兲关 兴 =1 4共⌫A兴关⌫A兲 = 1 4共⌫ A兴关⌫ A兲. 共31兲
This identity is sufficient to reproduce all possible Fierz identities by appropriately multiplying identity matrices by general matrices X , Y as 共X兲关Y兴 = 共X1兲关1Y兴 =1 4共X⌫CY兴关⌫ C兲 = 1 42 Tr关X⌫CY⌫D兴共⌫ D兴关⌫C兲. 共32兲
In particular, if X =⌫A and Y =⌫B, Eq.共32兲 leads to the Fierz identities
共⌫A兲关⌫B兴 = 1
42Tr关⌫
A⌫
C⌫B⌫D兴共⌫D兴关⌫C兲. 共33兲
The only remaining task is to calculate the expansion coef-ficients which are straightforward gamma matrix traces.15 The usual textbook Fierz identities 共see, for example, Ref. 11, p. 160兲 can be found when ⌫A
and⌫Bin Eq.共33兲 are chosen to form scalar quantities共under the full Lorentz trans-formations, including parity兲 such as 共␥兲关␥兴 or Eq. 共31兲 itself. An additional minus sign arises in the Fierz identities 共33兲 when we insert anticommuting fermion fields instead of numerical spinors.
IV. CHIRAL FIERZ IDENTITIES
The Fierz identities derived in Sec. III are not quite appro-priate when treating chirally projected combinations such as
共R␥兲关L␥
兴, 共34兲
where the two chiral projectors are R =12共1+␥5兲 and L=21共1 −␥5兲 because the expansion 共32兲 applied to Eq. 共34兲 still have some non-null coefficients to be calculated. For nonsca-lar combinations such as共R兲关R␥兴, the number of coeffi-cients to be calculated may become large. Moreover, the relatively simple form of certain chiral Fierz identities such as the form invariants16
共R␥兲关R␥
兴 = − 共R␥兴关R␥兲, 共35a兲 共L␥兲关L␥
兴 = − 共L␥兴关L␥兲, 共35b兲 suggests a simpler procedure should exist.
A better way to perform Fierz transformations for combi-nations such as Eq.共34兲 consists of rederiving Fierz identi-ties using a chiral basis
兵⌫A其 = 兵R,L,R␥
,L␥,其 共,= 0,1,2,3兲, 共36兲 where⬍, and its respective dual basis
兵⌫A其 =
兵
R,L,L␥,R␥,1
2
其
共,= 0,1,2,3兲, 共37兲 where⬍. Notice that because of the anticommuting na-ture of␥with␥5and projector properties, the dual of R␥is L␥. The orthogonality property between the bases共36兲 and 共37兲 is Tr关⌫A⌫ B兴 = 2␦ A B , 共38兲which implies the completeness relation 共 兲关 兴 =1 2共⌫A兴关⌫A兲 =12
兵
共R兴关R兲 + 共L兴关L兲 + 共R␥兴关L␥兲, + 共L␥兴关R␥兲 +共
12兴关
1 2兲其
, 共39兲where the extra 1 / 2 in the expansion is inserted to ac-count for the double ac-counting due to the implicit, sum-mation over all values and= −. Such a completeness relation directly leads to the chiral Fierz identities
共⌫A兲关⌫B兴 =1 4Tr关⌫
A⌫
C⌫B⌫D兴共⌫D兴关⌫C兲. 共40兲
To illustrate the usefulness of the chiral Fierz identities, we apply them to calculate the Fierz transform of the com-bination共34兲,
共R␥兲关L␥
兴 =14Tr关R␥⌫CL␥⌫D兴共⌫D兴关⌫C兲
=14Tr关R␥LL␥R兴共R兴关L兲 = 2共R兴关L兲, 共41兲 where the gamma matrix properties15 ␥␥= 4⫻1,
␥␣␥
= 0, and the trace cyclic property were used. More difficult examples can be worked out, for instance,
共R兲关R␥兴 =14Tr关RRR␥L␥兴共R␥兴关R兲 +41Tr
关
R 1 2␣R␥L␥兴
共R␥兴关
1 2␣兲
=32i共R␥兴关R兲 +21共R␥兴关R兲. 共42兲 Some labor can be saved by using the trace relationTr关R␣兴 = 2共g␣g− gg␣+ i␣兲, 共43a兲 Tr关L␣兴 = 2共g␣g− gg␣− i␣兲. 共43b兲 One can check the coefficients for the cases共兲=共␣兲 and 共␣兲=共0123兲.
Moreover, the combination of chiral Fierz identities 共40兲 with other completeness relations such as Eq. 共21兲 can be used to decompose four-quark operators carrying other quan-tum numbers like SU共3兲 color.2
The simplicity arises because only a few expansion coef-ficients are nonzero due to the projector properties of R / L and the commuting or anticommuting character of the bilin-ears with␥5. Equivalently, R / L projectors reduce the spinor vector space and the resulting projected spinors have to be the same on the two sides of the chiral Fierz identities.
V. SUMMARY
The well-known result that the Fierz identities are a con-sequence of the completeness of the Dirac bilinears as a basis
spanning the complex 4⫻4 complex matrices was reviewed. Recognizing that bases other than Dirac bilinears are equally possible permitted us to develop a better procedure to calcu-late the chiral Fierz identities by choosing chirally left/right projected matrices as a basis. The generality of the procedure was stressed and illustrated using the canonical basis of ma-trix vector spaces, which led to trivial relations in this case. The usefulness of Fierz-type relations depends on the par-ticular choice of basis and how nearly complete is the set of matrix objects共representations兲 of interest. The same unified framework was used to derive completeness relations for the generators of the SU共N兲 group in the fundamental represen-tation. Other matrix representations or other algebras can be analyzed as well, although they may not be complete and hence the corresponding Fierz-type identities may not be as useful as those presented here.
ACKNOWLEDGMENT
This work was supported by Conselho Nacional de Desen-volvimento Científico e Tecnológico 共CNPq兲.
a兲Electronic address: [email protected]
1M. Fierz, “Zur Fermischen Theorie des-Zerfalls,” Z. Phys. 104, 553–
565共1937兲.
2J. F. Donoghue, E. Golowich, and B. R. Holstein, Dynamics of the
Stan-dard Model共Cambridge University Press, 1994兲, pp. 217 and 221.
3J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics
共McGraw-Hill, New York, 1965兲.
4Y. Takahashi, “The Fierz identities,” in Progress in Quantum Field
Theory, edited by H. Ezawa and S. Kamefuchi共North-Holland,
Amster-dam, 1986兲, p. 121.
5J. F. Nieves and P. B. Pal, “Generalized Fierz identities,” Am. J. Phys.
72, 1100–1108共2004兲.
6R. Mohapatra and P. Pal, Massive Neutrinos in Physics and Astrophysics
共World Scientific, Singapore, 1991兲, pp. 165–70.
7The surprising implications and experimental evidences for neutrino
os-cillations can be found in W. C. Haxton and B. R. Holstein, “Neutrino physics,” Am. J. Phys. 68, 15–32共2000兲; also W. C. Haxton and B. R. Holstein, “Neutrino physics: An update,” Am. J. Phys. 72, 18–24共2004兲; see also M. C. Gonzalez-Garcia and Y. Nir, “Neutrino masses and mix-ing: Evidence and implications,” Rev. Mod. Phys. 75, 345–402共2003兲.
8See any linear algebra book, for example, I. M. Gel’fand, Lectures on
Linear Algebra共Interscience, New York, 1961兲.
9Any operation applied to an element in the vector space must result in
another element of the vector space.
10M
N共C兲 may be considered as a 2N2dimensional vector space if spanned
by N2real matrices and N2purely complex matrices with real expansion
coefficients only, that is, over the realsR.
11See C. Itzykson and J. B. Zuber, Quantum Field Theory共McGraw-Hill,
New York, 1980兲, p. 516, for an explicit representation of Gell-Mann matrices.
12
The SU共2兲 and SU共3兲 generators are兵12i其and兵 1
2a其, respectively. 13I will denote as Dirac bilinears the proper bilinears containing the two
spinors as well as the associated matrices alone, because the Fierz iden-tities do not depend on the spinors involved.
14The lowering of space-time indices is equivalent to the hermitian
conju-gation operation.
15See Ref. 11, Appendix.
16These identities were used to get from Eq.共1兲 to Eq. 共2兲 with a sign
difference due to the anticommutation of fermion fields.
1163 Am. J. Phys., Vol. 73, No. 12, December 2005 C. C. Nishi 1163
