The Simplest Form of the Lorentz Transformations
We report the simplest possible form to compute rotations around arbitrary axis and boosts in arbitrary directions for 4-vectors (space-time points, energy-momentum) and bi-vectors (electric and magnetic field vectors) by symplectic similarity transformations. The Lorentz transformations are based exclusively on real -matrices and require neither complex numbers nor special implementations of abstract entities like quaternions or Clifford numbers. No raising or lowering of indices is necessary. It is explained how the Lorentz transformations can be derived from the most simple second order Hamiltonian of general significance. Since this approach exclusively uses the real Clifford algebra , all calculations are based on real matrix algebra.
pacs:45.20.Jj, 05.45.Xt, 41.85.-p, 03.30.+p
The form of the Lorentz transformations (rotations and boosts) depends on the type of vectorial system used to represent space and time coordinates 111For the history of the different representations see Crowe ; Chappell .. The most common form of the Lorentz transformations however is unnecessarily complicated. We shall present here the simplest possible form and the simplest possible derivation. Our approach is similar to the space-time algebra (STA) suggested by David Hestenes GA ; STA . However, it differs from the latter insofar as we put more emphasis on the matrix representation of the Clifford algebra. This is physically significant since the Lorentz transformations can be derived from the second-order Hamiltonian of two classical coupled oscillators, which can generally be described by real matrices. Kim and Noz analyzed various forms in which the Pauli and Dirac algebra can be used to describe harmonic oscillators KimNoz .
Our approach originated from a practical problem in coupled charged particle optics, that suggested for its solution to survey all possible linear symplectic similarity transformations of two classical coupled degrees of freedom rdm_paper ; geo_paper . As a “by-product” we found that the Lorentz transformations can be re-interpreted as symplectic similarity transformations, i.e. canonical transformations as they are known in classical Hamiltonian mechanics. However, in this approach, physical observables like momentum and energy are not self-sufficient “fundamental” quantities. Instead they are (linear combinations of) the second moments of phase-space distributions (see Ref. (Goldstein ). In two essays we conjectured that this re-interpretation of the Lorentz transformations might also lead to a re-interpretation of quantum electrodynamics as a science of statistical moments in the phase space of spinors qed_paper ; osc_paper .
But no matter, if one is willing or able to follow our approach that far, the resulting formalism to derive the Lorentz transforms is extraordinarily simple and straightforward. Furthermore it provides a one-to-one relation between all terms that are possibly required to describe the coupling of two classical oscillators and the dynamics of a charged particle in an electromagnetic field.
In order to motivate our approach we describe the conventional vector formalism (CVF) first and contrast it with the suggested formalism of symplectic similarity transformations.
Ii Space described by Vectors
A position or direction in space is most commonly represented by vectors. As well-known, in CVF a “vector” is represented by a -matrix
or with the superscipt “T” for matrix transposition. Vectors are indicated by bold printed lower case letters, matrices by bold printed capital letters. If we construct unit vectors in each direction, then we may write:
The scalar product (dot product) of two vectors can be implemented as a product of a transposed -matrix times a -matrix
Unfortunately, this form to represent a vector has the undesired feature that the scalar multiplication changes the algebraic dimension and results - as the name suggests - in a scalar. Strange enough, there is a second type of vector multiplication, the so-called “vector” or “cross” product, which requires an extra symbol, namely the cross, and has its own definition:
At first sight the cross product is an oddity of -dimensional space and has no generalization to arbitrary dimensions and no obvious place within a generalized vector- and matrix-algebra. However, the cross product is physically and geometrically important and reflects physical properties of -dimensional spaces, namely the handedness of magnetic and gyroscopic forces. The need to define two different products indicates, that the unstructured “list” of coordinates does not adequately represent the structural properties of -dimensional physical space.
Let us consider the rotation of a vector by an angle about an arbitrary direction which is represented by a unit vector . The derivation of an appropriate formula requires the computation of the vector-components parallel and perpendicular to and it is helpful to use a drawing that clarifies the situation (see Fig. 1). Besides the and -function mainly vector addition and the computation of scalar and cross-products are needed in order to decompose the vector into the component parallel and perpendicular to , respectively.
From this we can derive the most simple formula of CVF, the formula of Rodriguez:
For the description of a supposedly fundamental operation like rotation in space, this formula is surprizingly complicated. But there is a much simpler formula available.
In the Hamiltonian Clifford Algebra that we suggest, the rotation of an arbitrary vector around an arbitrary direction is generated by a rotation matrix applied in the form of a similarity transformation to a matrix , namely:
where the rotation matrix is given by
in which has the same general structure as , namely that of a Hamiltonian matrix. The matrix depends on the rotation axis, but not on the vector to be rotated. All generators of rotations square to (i.e. are representations of the unit imaginary ), such that Eq. 9 yields Eulers formula:
Obviously the inverse transformation is given by the negative argument :
Also the CVF uses matrices to describe rotations. Since positions are represented in CVF by matrices, the rotation matrices , and have size and are no similarity transformation, but a multiplication of an orthogonal matrix and the vector
where indicates a rotation axis. These rotation matrices are
, and represent rotations around the coordinate axis , and . As already mentioned, the description of a general rotation of the vector around an arbitrary axis can be done by a single matrix multiplication with a matrix . This general matrix is far from being simple or intuitive. It is explicitely given in App. B.
Surprisingly enough, the conventional rotation matrices are not directly used to describe the motion of rigid bodies in -dimensional space. Instead, most textbooks suggest the use of Euler angles. The Euler angles are a powerful tool, but again are not simple or intuitive: Greiner for instance explains these angles with three figures Greiner . Even though the human mind is trained to grasp 3-dimensional situations, when it comes to real calculations, 3-dimensional space seems remarkably tedious. This is even worse when Lorentz boosts and electromagnetic fields are considered.
ii.2 Lorentz Boost of -vectors
Jacksons “Electrodynamics” presents the following formula, with the restriction that the boost must be along Jackson :
and, for the general case:
Again it is required to split vectors into the parallel and perpendicular components.
In the Hamiltonian Clifford Algebra that we suggest, the boost of an arbitrary 4-vector is performed by a boost matrix in the form of a symplectic similarity transformation , namely:
where the boost matrix is given by
in which has the same structure as , namely that it is a Hamiltonian matrix. Generators of boosts squares to , such that
Again the inverse matrix is given by the negative argument . This means that there is no significant formal difference between rotations and boosts.
ii.3 Lorentz Boost of Electromagnetic Fields
So far our treatment concerned only the transformations of “vector” components. Now we include electromagnetic fields. The corresponding formulas are, again assumed that the parallel and perpendicular components are computed beforehand Jackson :
In the Hamiltonian Clifford Algebra that we suggest, the boost of an arbitrary 4-vector together with an arbitrary electromagnetic field, is represented by a boost matrix in the form of a symplectic similarity transformation , namely:
where the boost matrix is given by
in which has the same structure as , namely that it is a Hamiltonian matrix. This structure can be derived from a classical bilinear Hamiltonian. The matrices contain exactly the required number of independent parameters, namely ten, to represent a vector and field components, the latter being naturally grouped into sets of components. The use of complex numbers is not required. The combination of a simultaneous boost and rotation is, due to the “superposition principle”, obtained as the matrix exponential of the sum of the generators:
The composition of the generators is simple and can be derived in a straight forward manner from the structure of a symplectic Hamiltonian Clifford algebra , which represents a complete set of -matrices and is a real-valued variant of the Dirac algebra.
Iii Matrix Representations
Let us have a second look at (Eq. 2), at two vectors and their product
If we consider to be usual (commutative) -vectors, we obtain
with the Kronecker 222 The Kronecker delta is defined by: for and for . the well-known expression of the scalar product, since all mixed terms in Eq. LABEL:eq1 vanish. Now let us consider the case that the unit elements are not represented by (column) vectors, but by (real) square matrices. In this case, the elements and do not commute in the general case. If we consider the very opposite of commutation, namely that they pairwise anti-commute, then
so that, since Eq. 26 implies that , one finds
where the bold-face represents a unit matrix. In this case the resulting expressions are obviously a combination of the scalar and the vector product. This becomes clearer, if we identify the products
with a new type of (unit-)vector, a “bi-vector”. Since the vector elements anti-commute and square to , the elements of the bi-vector square to :
and (as can easily be proved) they pairwise anti-commute, just as we presumed for the vector-type elements 333 Note that the bi-vector is a representation of the quaternion elements , and . .
As mentioned before, one major advantage of the representation of spatial unit vectors by real square matrices is, that all sums and products of matrices are again square matrices of the same dimension. It is therefore possible to compute arbitrary analytical functions of matrices in the form of Taylor series, for instance the matrix exponential. While computation of the matrix exponential of is - in the general case - quite involved exp_paper , it significantly simplifies, if the argument squares to the (positive or negative) unit matrix . The Taylor series is then splitted into the even and odd two partial series, such that with (with the sign ) one obtains:
such that with one finds
If the matrix would square to the positive unit matrix, i.e. if it follows that
Obviously we have . Consider the transformation of a “vector” according to
If the transformation matrix commutes with , then this component is unchanged. But what happens, if it does not commute?
iii.1 Rotations by Matrices
Let us explicitely calculate the result of the transformation (Eq. 33) for , where we use the abbreviations , , and :
where , which we evaluate component-wise:
Now, the anti-commutation rules yield:
such that with and :
For the -component one obtains equivalently
while the -component is unchanged since commutes with . In summary we obtain a rotation around the -axis:
Hence, if such anti-commuting “unit”-matrices exist, then they can be used to represent spatial rotations.
iii.2 Clifford Algebras
Up to now we did not specify the exact form of the matrices - we only assumed that they anti-commute and square to the unit matrix. This means that the exact form of the matrices is not essential for the purpose of representing rotations. This fact is often interpreted in such a way, that the elements do not have to be represented by matrices at all. Instead it is often suggested to regard as abstract elements of a so-called Clifford algebra (CA). This view is possible and legitimate, but then the connection with the Hamiltonian formalism (and therefore some physical insight) get’s lost.
A Clifford algebra that is generated by three elements , and with positive norm (), is named . More generally speaking a Clifford algebra has pairwise anti-commuting generators, of which square to and square to . From combinatorics one finds that has -vectors and in summary it has
linear independent elements, where the -vector is the scalar (unit element)
and the -vector, i.e. the product of all generators,
is the so-called pseudo-scalar.
has linear independent elements, namely the generators, bi-vectors,
the scalar and the pseudoscalar
(or , respectively).
This means, that there is no one-to-one isomorphism to a specific real square matrix of size .
However, up to now, we did not yet include a time coordinate. In order to represent a coordinate
in Minkowski space-time, we need vector-type elements and therefore we introduce another
unit element, which might be called or . In this case we have and hence
linear independent elements, a size that would perfectly match to
-matrices 444As a result known from representation theory, there are some restrictive conditions for the
representation of Clifford algebras by real matrices, namely that either or
with arbitary integer , often written as
A possible choice for the real -matrices is given by555For better readability the zeros are replaced by dots.:
From these “generators”, which mutually anti-commute, the following bi-vectors are obtained my matrix multiplication:
Hence the matrices , and represent the bi-vector of Eq. 28. Since the new generator anti-commutes with , and , it commutes with , and and is hence unchanged by the rotations generated by (the matrix exponential of) these bi-vectors. It is therefore no spatial coordinate. Furthermore we have more bi-vectors , and , which square to :
From Eq. 32 we know that , and can not generate rotations. What do they generate then? As generates rotations in the -plane, the bi-vector generates a transformation in the plane of and . Such transformations are called boosts.
iii.3 Boosts by Matrices
We now examine the result of the transformation of a “vector” according to:
where . Obviously the product commutes
with and , so that and . For the other two components
we evaluate component-wise 666
Given an arbitrary matrix that is an unknown vector.
Since the trace of all Dirac matrices vanishes except for the unit matrix,
one obtains the coefficient of by the formula
where with and , we used the following theorems
If we use the conventional notation and (i.e. ), then we obtain the Lorentz boost along the -axis
where is the so-called “rapidity”.
Thus we have demonstrated that a -vector in Minkowski space-time can be represented by matrices and that both, rotations and boosts of -vectors can be written as similarity transformations. Next we prove that rotations and boosts of electromagnetic fields follow the exact same approach, i.e. can be represented by exactly the same similarity transformations, if the fields are “encoded” as bi-vectors:
iii.4 Rotations and Boosts of Electromagnetic fields
Again we use a rotation around the -axis (see Eq. 34), i.e. the generator is and it commutes with , which is trivial and with , which is also quickly verified. But anti-commutes with and , so that:
The electric field components in the -plane are (with and , and ):
The terms of the magnetic field transform in exactly the same way:
A boost along is generated by , which commutes with itself and with , such that the electromagnetic field components in the direction of the boost are unchanged. The electric field components in the plane perpendicular to the boost are (with and , and ):
With and we obtain:
With and we obtain:
such that (again with and ):
These equations are in exact agreement with the Lorentz transformation of the electromagnetic fields.
iii.5 The Lorentz Force
Hence we obtain a perfectly simple and systematic approach not only of rotations but also of boosts, if we associate the -vector components with (time-like, energy ) and , and for the space-like components (momentum, ) and furthermore associate the electromagnetic fields with the bi-vectors 777 This mapping has been called electro-mechanical equivalence (EMEQ) rdm_paper ; geo_paper . :
This mapping has also physical significance firstly, because magnetic fields really act as generators of rotational motion as well as electric field act as generators of boosts, but secondly, with the introduction of an appropriate weight factor , the Lorentz force can be written as rdm_paper ; geo_paper :
where the overdot indicates the derivative with respect to proper time. The evaluation of the components gives, written in conventional vector form:
with this becomes (with ):
To summarize: if we make use of elements (out of ) of the Clifford algebra , we find a systematic description of minimal complexity for a massive particle in an (“external”) electromagnetic field - simply by the use of -matrices instead of the conventional vector-notation. How is this possible and what about the remaining elements of the complete Clifford algebra?
iii.6 The Remaining Matrices
The remaining matrices are not directly used, but are given to complete the list of real -matrices: