In mathematical physics, the gamma matrices, also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin particles. Gamma matrices were introduced by Paul Dirac in 1928.[1][2]
In Dirac representation, the four contravariant gamma matrices are
is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly, and where denotes the Kronecker product and the (for j = 1, 2, 3) denote the Pauli matrices.
In addition, for discussions of group theory the identity matrix (I) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices
The "fifth matrix" is not a proper member of the main set of four; it is used for separating nominal left and right chiral representations.
The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacetime dimensions, the four gammas, above, together with the fifth gamma-matrix to be presented below generate the Clifford algebra.
Mathematical structure
editThe defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
where the curly brackets represent the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix.
This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. Covariant gamma matrices are defined by
and Einstein notation is assumed.
Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation:
or a multiplication of all gamma matrices by , which of course changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by
Physical structure
editThe Clifford algebra over spacetime V can be regarded as the set of real linear operators from V to itself, End(V), or more generally, when complexified to as the set of linear operators from any four-dimensional complex vector space to itself. More simply, given a basis for V, is just the set of all 4×4 complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric ημν. A space of bispinors, Ux , is also assumed at every point in spacetime, endowed with the bispinor representation of the Lorentz group. The bispinor fields Ψ of the Dirac equations, evaluated at any point x in spacetime, are elements of Ux (see below). The Clifford algebra is assumed to act on Ux as well (by matrix multiplication with column vectors Ψ(x) in Ux for all x). This will be the primary view of elements of in this section.
For each linear transformation S of Ux, there is a transformation of End(Ux) given by S E S−1 for E in If S belongs to a representation of the Lorentz group, then the induced action E ↦ S E S−1 will also belong to a representation of the Lorentz group, see Representation theory of the Lorentz group.
If S(Λ) is the bispinor representation acting on Ux of an arbitrary Lorentz transformation Λ in the standard (4 vector) representation acting on V, then there is a corresponding operator on given by equation:
showing that the quantity of γμ can be viewed as a basis of a representation space of the 4 vector representation of the Lorentz group sitting inside the Clifford algebra. The last identity can be recognized as the defining relationship for matrices belonging to an indefinite orthogonal group, which is written in indexed notation. This means that quantities of the form
should be treated as 4 vectors in manipulations. It also means that indices can be raised and lowered on the γ using the metric ημν as with any 4 vector. The notation is called the Feynman slash notation. The slash operation maps the basis eμ of V, or any 4 dimensional vector space, to basis vectors γμ. The transformation rule for slashed quantities is simply
One should note that this is different from the transformation rule for the γμ, which are now treated as (fixed) basis vectors. The designation of the 4 tuple as a 4 vector sometimes found in the literature is thus a slight misnomer. The latter transformation corresponds to an active transformation of the components of a slashed quantity in terms of the basis γμ, and the former to a passive transformation of the basis γμ itself.
The elements form a representation of the Lie algebra of the Lorentz group. This is a spin representation. When these matrices, and linear combinations of them, are exponentiated, they are bispinor representations of the Lorentz group, e.g., the S(Λ) of above are of this form. The 6 dimensional space the σμν span is the representation space of a tensor representation of the Lorentz group. For the higher order elements of the Clifford algebra in general and their transformation rules, see the article Dirac algebra. The spin representation of the Lorentz group is encoded in the spin group Spin(1, 3) (for real, uncharged spinors) and in the complexified spin group Spin(1, 3) for charged (Dirac) spinors.
Expressing the Dirac equation
editIn natural units, the Dirac equation may be written as
where is a Dirac spinor.
Switching to Feynman notation, the Dirac equation is
The fifth "gamma" matrix, γ5
editIt is useful to define a product of the four gamma matrices as , so that
- (in the Dirac basis).
Although uses the letter gamma, it is not one of the gamma matrices of The index number 5 is a relic of old notation: used to be called " ".
has also an alternative form:
using the convention or
using the convention Proof:
This can be seen by exploiting the fact that all the four gamma matrices anticommute, so
where is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If denotes the Levi-Civita symbol in n dimensions, we can use the identity . Then we get, using the convention
This matrix is useful in discussions of quantum mechanical chirality. For example, a Dirac field can be projected onto its left-handed and right-handed components by:
Some properties are:
- It is Hermitian:
- Its eigenvalues are ±1, because:
- It anticommutes with the four gamma matrices:
In fact, and are eigenvectors of since
- and
Five dimensions
editThe Clifford algebra in odd dimensions behaves like two copies of the Clifford algebra of one less dimension, a left copy and a right copy.[3]: 68 Thus, one can employ a bit of a trick to repurpose i γ 5 as one of the generators of the Clifford algebra in five dimensions. In this case, the set {γ 0, γ 1, γ 2, γ 3, i γ 5} therefore, by the last two properties (keeping in mind that i 2 ≡ −1) and those of the ‘old’ gammas, forms the basis of the Clifford algebra in 5 spacetime dimensions for the metric signature (1,4).[a] .[4]: 97 In metric signature (4,1), the set {γ 0, γ 1, γ 2, γ 3, γ 5} is used, where the γ μ are the appropriate ones for the (3,1) signature.[5] This pattern is repeated for spacetime dimension 2n even and the next odd dimension 2n + 1 for all n ≥ 1.[6]: 457 For more detail, see higher-dimensional gamma matrices.
Identities
editThe following identities follow from the fundamental anticommutation relation, so they hold in any basis (although the last one depends on the sign choice for ).
Miscellaneous identities
edit1.
Proof | ||||||||||||||||||
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Take the standard anticommutation relation: One can make this situation look similar by using the metric :
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2.
Proof |
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Similarly to the proof of 1, again beginning with the standard commutation relation: |
3.
Proof | ||||||||||||
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To show Use the anticommutator to shift to the right
Using the relation we can contract the last two gammas, and get
Finally using the anticommutator identity, we get |
4.
Proof | ||||||||||
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5.
Proof |
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If then and it is easy to verify the identity. That is the case also when , or . On the other hand, if all three indices are different, , and and both sides are completely antisymmetric; the left hand side because of the anticommutativity of the matrices, and on the right hand side because of the antisymmetry of . It thus suffices to verify the identities for the cases of , , and . |
6. where
Proof | ||||||||||
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For and both sides vanish. Otherwise, multiplying identity 5 by from the right gives that
where since . The left hand side of this equation also vanishes since by property 3. Rearranging gives that
Note that for (for , vanishes) by the standard anticommutation relation. It follows that
Multiplying from the left times and using that yields the desired result. |
Trace identities
editThe gamma matrices obey the following trace identities:
- Trace of any product of an odd number of is zero
- Trace of times a product of an odd number of is still zero
Proving the above involves the use of three main properties of the trace operator:
- tr(A + B) = tr(A) + tr(B)
- tr(rA) = r tr(A)
- tr(ABC) = tr(CAB) = tr(BCA)
Proof of 1 | ||||
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From the definition of the gamma matrices, We get or equivalently, where is a number, and is a matrix.
This implies |
Proof of 2 | ||||||||
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To show First note that We'll also use two facts about the fifth gamma matrix that says: So lets use these two facts to prove this identity for the first non-trivial case: the trace of three gamma matrices. Step one is to put in one pair of 's in front of the three original 's, and step two is to swap the matrix back to the original position, after making use of the cyclicity of the trace.
This can only be fulfilled if The extension to 2n + 1 (n integer) gamma matrices, is found by placing two gamma-5s after (say) the 2n-th gamma-matrix in the trace, commuting one out to the right (giving a minus sign) and commuting the other gamma-5 2n steps out to the left [with sign change (-1)^2n = 1]. Then we use cyclic identity to get the two gamma-5s together, and hence they square to identity, leaving us with the trace equalling minus itself, i.e. 0. |
Proof of 3 |
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If an odd number of gamma matrices appear in a trace followed by , our goal is to move from the right side to the left. This will leave the trace invariant by the cyclic property. In order to do this move, we must anticommute it with all of the other gamma matrices. This means that we anticommute it an odd number of times and pick up a minus sign. A trace equal to the negative of itself must be zero. |
Proof of 4 | ||||||
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To show Begin with, |
Proof of 5 | ||||||||||||
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For the term on the right, we'll continue the pattern of swapping with its neighbor to the left, Again, for the term on the right swap with its neighbor to the left, Eq (3) is the term on the right of eq (2), and eq (2) is the term on the right of eq (1). We'll also use identity number 3 to simplify terms like so: So finally Eq (1), when you plug all this information in gives The terms inside the trace can be cycled, so So really (4) is or |
Proof of 6 | ||||||||||||
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To show
begin with
Add to both sides of the above to see
Now, this pattern can also be used to show
Simply add two factors of , with different from and . Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace. So,
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Proof of 7 |
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For a proof of identity 7, the same trick still works unless is some permutation of (0123), so that all 4 gammas appear. The anticommutation rules imply that interchanging two of the indices changes the sign of the trace, so must be proportional to . The proportionality constant is , as can be checked by plugging in , writing out , and remembering that the trace of the identity is 4. |
Proof of 8 | |||||||||||||||||
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Denote the product of gamma matrices by Consider the Hermitian conjugate of :
Conjugating with one more time to get rid of the two s that are there, we see that is the reverse of . Now,
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Normalization
editThe gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose
- , compatible with
and for the other gamma matrices (for k = 1, 2, 3)
- , compatible with
One checks immediately that these hermiticity relations hold for the Dirac representation.
The above conditions can be combined in the relation
The hermiticity conditions are not invariant under the action of a Lorentz transformation because is not necessarily a unitary transformation due to the non-compactness of the Lorentz group.[citation needed]
Charge conjugation
editThe charge conjugation operator, in any basis, may be defined as
where denotes the matrix transpose. The explicit form that takes is dependent on the specific representation chosen for the gamma matrices, up to an arbitrary phase factor. This is because although charge conjugation is an automorphism of the gamma group, it is not an inner automorphism (of the group). Conjugating matrices can be found, but they are representation-dependent.
Representation-independent identities include:
The charge conjugation operator is also unitary , while for it also holds that for any representation. Given a representation of gamma matrices, the arbitrary phase factor for the charge conjugation operator can also be chosen such that , as is the case for the four representations given below (Dirac, Majorana and both chiral variants).
Feynman slash notation
editThe Feynman slash notation is defined by
for any 4-vector .
Here are some similar identities to the ones above, but involving slash notation:
- [7]
- [7]
- [7]
- where is the Levi-Civita symbol and Actually traces of products of odd number of is zero and thus
- for n odd.[8]
Many follow directly from expanding out the slash notation and contracting expressions of the form with the appropriate identity in terms of gamma matrices.
Other representations
editThe matrices are also sometimes written using the 2×2 identity matrix, , and
where k runs from 1 to 3 and the σk are Pauli matrices.
Dirac basis
editThe gamma matrices we have written so far are appropriate for acting on Dirac spinors written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:
In the Dirac basis, the charge conjugation operator is real antisymmetric,[9]: 691–700
Weyl (chiral) basis
editAnother common choice is the Weyl or chiral basis, in which remains the same but is different, and so is also different, and diagonal,
or in more compact notation:
The Weyl basis has the advantage that its chiral projections take a simple form,
The idempotence of the chiral projections is manifest.
By slightly abusing the notation and reusing the symbols we can then identify
where now and are left-handed and right-handed two-component Weyl spinors.
The charge conjugation operator in this basis is real antisymmetric,
The Dirac basis can be obtained from the Weyl basis as
via the unitary transform
Weyl (chiral) basis (alternate form)
editAnother possible choice[10] of the Weyl basis has
The chiral projections take a slightly different form from the other Weyl choice,
In other words,
where and are the left-handed and right-handed two-component Weyl spinors, as before.
The charge conjugation operator in this basis is
This basis can be obtained from the Dirac basis above as via the unitary transform
Majorana basis
editThere is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Regarding the Pauli matrices, the basis can be written as
where is the charge conjugation matrix, which matches the Dirac version defined above.
The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric (+, −, −, −), in which squared masses are positive. The Majorana representation, however, is real. One can factor out the to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the is that the only possible metric with real gamma matrices is (−, +, +, +).
The Majorana basis can be obtained from the Dirac basis above as via the unitary transform
Cl1,3(C) and Cl1,3(R)
editThis article may benefit from being shortened by the use of summary style. |
The Dirac algebra can be regarded as a complexification of the real algebra Cl1,3( ), called the space time algebra:
Cl1,3( ) differs from Cl1,3( ): in Cl1,3( ) only real linear combinations of the gamma matrices and their products are allowed.
Two things deserve to be pointed out. As Clifford algebras, Cl1,3( ) and Cl4( ) are isomorphic, see classification of Clifford algebras. The reason is that the underlying signature of the spacetime metric loses its signature (1,3) upon passing to the complexification. However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" (at the very least impractical) since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.
Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to −1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.[11]: x–xi
In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Clp,q( ) for arbitrary dimensions p,q. The Weyl spinors transform under the action of the spin group . The complexification of the spin group, called the spinc group , is a product of the spin group with the circle The product just a notational device to identify with The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the component, which can be identified with the fiber of the electromagnetic interaction. The is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor (Eigenspinoren des Ladungskonjugationsoperators), which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the part coming from the complexification. The ELKO spinor is a Lounesto class 5 spinor.[12]: 84
However, in contemporary practice in physics, the Dirac algebra rather than the space-time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.
Other representation-free properties
editThe gamma matrices are diagonalizable with eigenvalues for , and eigenvalues for .
Proof |
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This can be demonstrated for and follows similarly for . We can rewrite as By a well-known result in linear algebra, this means there is a basis in which is diagonal with eigenvalues . |
In particular, this implies that is simultaneously Hermitian and unitary, while the are simultaneously anti–Hermitian and unitary.
Further, the multiplicity of each eigenvalue is two.
Proof |
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If is an eigenvector of then is an eigenvector with the opposite eigenvalue. Then eigenvectors can be paired off if they are related by multiplication by Result follows similarly for |
More generally, if is not null, a similar result holds. For concreteness, we restrict to the positive norm case with The negative case follows similarly.
Proof |
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It can be shown so by the same argument as the first result, is diagonalizable with eigenvalues We can adapt the argument for the second result slightly. We pick a non-null vector which is orthogonal to Then eigenvectors can be paired off similarly if they are related by multiplication by |
It follows that the solution space to (that is, the kernel of the left-hand side) has dimension 2. This means the solution space for plane wave solutions to Dirac's equation has dimension 2.
This result still holds for the massless Dirac equation. In other words, if null, then has nullity 2.
Proof |
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If null, then By generalized eigenvalue decomposition, this can be written in some basis as diagonal in Jordan blocks with eigenvalue 0, with either 0, 1, or 2 blocks, and other diagonal entries zero. It turns out to be the 2 block case. The zero case is not possible as if by linear independence of the we must have But null vectors are by definition non-zero. Consider and a zero-eigenvector of . Note is also null and satisfies If , then it cannot simultaneously be a zero eigenvector of by (*). Considering , if we apply then we get . Therefore, after a rescaling, and give a Jordan block. This gives a pairing. There must be another zero eigenvector of
There is also a pleasant structure to these pairs. If left arrows correspond to application of , and right arrows to application of , and is a zero eigenvector of , up to scalar factors we have
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Euclidean Dirac matrices
editIn quantum field theory one can Wick rotate the time axis to transit from Minkowski space to Euclidean space. This is particularly useful in some renormalization procedures as well as lattice gauge theory. In Euclidean space, there are two commonly used representations of Dirac matrices:
Chiral representation
editNotice that the factors of have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra
will emerge. It is also worth noting that there are variants of this which insert instead on one of the matrices, such as in lattice QCD codes which use the chiral basis.
In Euclidean space,
Using the anti-commutator and noting that in Euclidean space , one shows that
In chiral basis in Euclidean space,
which is unchanged from its Minkowski version.
Non-relativistic representation
editFootnotes
edit- ^ The set of matrices (Γa) = (γ μ, i γ 5 ) with a = (0, 1, 2, 3, 4) satisfy the five-dimensional Clifford algebra {Γa, Γb} = 2 ηab
See also
editCitations
edit- ^ Kukin 2016.
- ^ Lonigro 2023.
- ^ Jost 2002.
- ^ Tong 2007, These introductory quantum field theory notes are for Part III (masters level) students..
- ^ Weinberg 2002, § 5.5.
- ^ de Wit & Smith 2012.
- ^ a b c Feynman, Richard P. (1949). "Space-time approach to quantum electrodynamics". Physical Review. 76 (6): 769–789. doi:10.1103/PhysRev.76.769 – via APS.
- ^ Kaplunovsky 2008.
- ^ Itzykson & Zuber 2012.
- ^ Kaku 1993.
- ^ Hestenes 2015.
- ^ Rodrigues & Oliveira 2007.
References
edit- de Wit, B.; Smith, J. (2 December 2012). Field Theory in Particle Physics, Volume 1. Elsevier. ISBN 978-0-444-59622-2.[1]
- Halzen, Francis; Martin, Alan D. (17 May 2008). Quark & Leptons: An Introductory Course in Modern Particle Physics. Wiley India Pvt. Limited. ISBN 978-81-265-1656-8.
- Hestenes, David (2015). Space-Time Algebra. Springer International Publishing. ISBN 978-3-319-18412-8.
- Itzykson, Claude; Zuber, Jean-Bernard (20 September 2012). Quantum Field Theory. Courier Corporation. Appendix A. ISBN 978-0-486-13469-7.
- Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Springer. p. 68, Corollary 1.8.1. ISBN 978-3-540-42627-1.
- Kaku, Michio (1993). Quantum Field Theory: A Modern Introduction. Oxford University Press. ISBN 978-0-19-509158-8.
- Kaplunovsky, Vadim (2008). "Traceology" (PDF). Quantum Field Theory (course homework / class notes). Physics Department. University of Texas at Austin. Archived from the original (PDF) on 2019-11-13.
- Kukin, V.D. (2016). "Dirac matrices - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2023-11-02.
- Lonigro, Davide (2023). "Dimensional reduction of the Dirac equation in arbitrary spatial dimensions". The European Physical Journal Plus. 138 (4): 324. arXiv:2212.11965. Bibcode:2023EPJP..138..324L. doi:10.1140/epjp/s13360-023-03919-0.
- Pauli, W. (1936). "Contributions mathématiques à la théorie des matrices de Dirac". Annales de l'Institut Henri Poincaré. 6: 109.
- Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview Press. chapter 3.2. ISBN 0-201-50397-2.
- Rodrigues, Waldyr A.; Oliveira, Edmundo C. de (2007). The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach. Springer Science & Business Media. ISBN 978-3-540-71292-3.
- Tong, David (2007). Lectures on Quantum Field Theory (course lecture notes). David Tong at University of Cambridge. p. 93. Retrieved 2015-03-07.
- Weinberg, S. (2002). The Quantum Theory of Fields. Vol. 1. Cambridge University Press. ISBN 0-521-55001-7 – via Internet Archive (archive.org).
- Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton, NJ: Princeton University Press. chapter II.1. ISBN 0-691-01019-6.
External links
edit- Dirac matrices on mathworld including their group properties
- Dirac matrices as an abstract group on GroupNames
- "Dirac matrices", Encyclopedia of Mathematics, EMS Press, 2001 [1994]