- Email: [email protected]
A new perspective on spacetime 4D rotations and the SO(4) transformation group
Results in Physics 13 (2019) 102141
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
A new perspective on spacetime 4D rotations and the SO(4) transformation group
T
Mikołaj Myszkowski Borówiec Pod Borem 25, Poznań 62-023, Poland
A R T I C LE I N FO
A B S T R A C T
Keywords: 4D rotations Lorentz transformation Noether’s theorem
In quantum field theory, every conservation law is enforced by adequate symmetries. The Lorentz group, which represents two continuous symmetries: rotations in 3D Euclidean space and Lorentz boosts, generates conservation of the angular momentum tensor. This article shows that both Lorentz boosts and 3D rotations can be replaced by spatio-temporal rotations in Minkowski spacetime. Noether’s theorem certifies that the consequences of 4D rotations are identical to the Lorentz transformation group SO(4). I demonstrate how to build a spatio-temporal rotation matrix that preserves the spacetime interval, and that the resulting matrix represents the special orthogonal group of order 4 (SO(4)).
1. Introduction Symmetries play the main role in modern physics [1–3]. As long as the world was described by classical theories (non-quantum ones), miscellaneous laws of physics were added ad hoc, as a theory paradigm. When quantum mechanics was finally formulated, Noether’s theorem indicated that every symmetry forms a conservation law [1,4–6]. One of the oldest and easiest symmetries are 3D rotations. If the Lagrangian of a system L preserves its form under that transformation, it results in the emergence of conserved Noether currents and Noether charges, i.e. a 3D angular momentum tensor [5]. Transformations of Lorentz group SO(4) [7,8] (a special orthogonal group of order 4, also considered as a 4D Euclidean rotational group) can be represented in a 4 × 4 matrix form, where spatial components are common rotations and temporal components are Lorentz transformations [9,1,5]. In this article I will show that new 4D spacetime rotations express a Lorentz transformation. In order to preserve a spacetime interval (in the 3 dimensional case, rotations must preserve a vector norm, in a non-Euclidean 4D space rotations must preserve a four-vector norm, determined by a metric tensor), a complex analysis must be involved. In the subsequent calculations it is assumed that a metric tensor in flat spacetime is:
0 0 ⎤ ⎡1 0 0 −1 0 0 ⎥ gμν = ⎢ ⎢0 0 − 1 0 ⎥ 0 − 1⎦ ⎣0 0
(1)
Lorentz boosts, unlike 3D rotations, influence time as well [10–13]. This implies the association with time. These two transformations assembled together can be displayed as the result of multiplications.
Because Lorentz boosts and 3D rotations preserve a spacetime interval, the result of a complete transformation needs to preserve this quantity too. 2. Spatio-temporal (4D) rotation matrix To construct a full 4D Euclidean rotation matrix, a minimum of six 4 × 4 matrixes [14] (each represents a different rotation) need to be multiplied. It is worth mentiontioning that there are already some similarities with a SO(4) Lorentz group (Table 1): It is convenient to assign different angles to adequate rotations (Table 2): Matrices R (α ), R (β ), R (γ ) are the same as in a SO(3) rotation group case [15,16], hence:
⎡1 0 R (α ) = ⎢ ⎢0 ⎢ ⎣0
0 0 0 ⎤ 1 0 0 ⎥ , 0 cosα − sinα ⎥ ⎥ 0 sinα cosα ⎦
0 ⎡1 0 0 cosγ − sinγ R (γ ) = ⎢ ⎢ 0 sinγ cosγ ⎢ 0 ⎣0 0
0 ⎡1 ⎢ 0 cosβ R (β ) = ⎢ 0 0 ⎢ 0 − sinβ ⎣
0⎤ 0⎥ 0⎥ ⎥ 1⎦
0 0 ⎤ 0 sinβ ⎥ 1 0 ⎥ 0 cosβ ⎥ ⎦
(2)
(3)
However, the rest of the matrices take a less obvious form. Let’s assume that all components of R (λ ), R (θ), R (ϕ) are different from R (α ), R (β ), R (γ ) only by a multiplicative constant (as it will emerge later, this is a complex constant):
E-mail address: [email protected]. https://doi.org/10.1016/j.rinp.2019.02.077 Received 3 September 2018; Received in revised form 12 February 2019; Accepted 23 February 2019 Available online 26 February 2019 2211-3797/ © 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Results in Physics 13 (2019) 102141
M. Myszkowski
r00 = cosϕcosθcosλ r01 = ∓ icosϕcosθsinλ r02 = ∓ isinθcosϕ r03 = ∓ isinϕ r10 = ± icosβ (sinγ sinθcosλ − cosγ sinλ ) ∓ isinβ sinϕcosθ cosλ r11 = cosβ (cosγ cosλ + sinγ sinθsinλ ) − sinβ sinϕcosθsinλ r12 = −sinβ sinθsinϕ − cosβ sinγ cosθ r13 = sinβ cosϕ r20 = ∓ icosα (sinγ sinλ + cosγ sinθcosλ ) ∓ isinα (−sinβ (sinγ sinθcosλ − cosγ sinλ ) − cosβ sinϕcosθ cosλ ) r21 = cosα (sinγ cosλ − cosγ sinθsinλ )− − sinα (−sinβ (cosγ cosλ + sinγ sinθsinλ ) − cosβ sinϕcosθsinλ ) r22 = cosα cosγ cosθ + sinα ( −sinβ sinγ cosθ + cosβsinθsinϕ) r23 = −sinα cosβ cosϕ r30 = ∓ isinα (sinγ sinλ + cosγ sinθcosλ ) ± icosα ( −sinβ (sinγ sinθ cosλ − cosγ sinλ ) − cosβ sinϕcosθcosλ ) r31 = sinα (sinγ cosλ − cosγ sinθsinλ )− + cosα ( −sinβ (cosγ cosλ + sinγ sinθsinλ ) − cosβ sinϕcosθsinλ ) r32 = sinα cosγ cosθ − cosα (−sinβ sinγ cosθ + cosβ sinθsinϕ) r33 = cosα cosβ cosϕ
Table 1 Similarities of SO(4) group and 4D rotations. SO(4) group transformations
Spatio-temporal 4D rotations
6 independent parameters: 3 velocity components and 3 angles
6 independent parameters: 3 spatial angles and 3 “spatio-temporal” angles
Table 2 Adequate rotations and assigned angles. α t-x
Angle: Rotation plane
β t-y
⎡ a11cosλ − a12sinλ a21sinλ a22cosλ R (λ ) = ⎢ ⎢ 0 0 ⎢ 0 ⎣ 0
0 0 1 0
0⎤ 0 ⎥, 0⎥ ⎥ 1⎦
γ t-z
λ y-z
ϕ x-y
θ x-z
⎡ a11cosθ 0 R (θ) = ⎢ ⎢ a21sinθ ⎢ ⎣ 0
0 − a12sinθ 1 0 0 a22cosθ 0 0
0⎤ 0⎥ 0⎥ ⎥ 1⎦ (4)
⎡ a11cosϕ 0 R (ϕ) = ⎢ ⎢ 0 ⎢ a sinϕ ⎣ 21
0 1 0 0
0 − a12sinϕ ⎤ 0 0 ⎥ ⎥ 1 0 0 a22cosϕ ⎥ ⎦
3. Similarities of 4D rotations and Lorentz boosts
Using a norm invariance condition for the R (λ ) transformation [11,12], (ds′2 = ds 2 ) I obtain (please note that the matrix composed of t , x , y, z is represented by a row vector):
In order to find a condition which makes a spatio-temporal rotation identical to a Lorentz boost, one should demand R (λ ) = Λ(ν1) . It emerges that cosλ = γ1 and ∓ isinλ = −γ1ν1. Thus:
(ds′)2 = (t ′)2 − (x ′)2 = (a11 t cosλ + a21 x sinλ )2 − (−a12 t sinλ + a22 x cosλ )2 2 2 2 2 2 2 = a11 t cos2 λ + 2a11 a21 xt cosλsinλ + a21 x sin2 λ − a12 t sin2 λ 2 2 + 2a12 a22 xt cosλ sinλ − a22 x cos2 λ = t 2 − x 2
tanλ =
(6)
Therefore, multiplicative components need to comply with the undermentioned conditions:
λR = tanh−1 (∓ ν1) = ln
In order to decrease the number of possibilities, the conditions R (λ = 0) = , R (θ = 0) = , R (ϕ = 0) = are imposed, ( is identity matrix) which give a11 = a22 = 1. After summing up, there is still a minuscule freedom of choice (because a11 = a22 = 1, a12 = −a21 = ± i ). Because this freedom should not affect further considerations, the following representation can be chosen:
0 0 1 0
0⎤ 0 ⎥, 0⎥ 1⎥ ⎦
⎡ cosθ 0 R (θ) = ⎢ ⎢ ∓ isinθ ⎢ 0 ⎣
0 ∓ isinθ 1 0 0 cosθ 0 0
0 1 0 0
0 ∓ isinϕ ⎤ 0 0 ⎥ 1 0 ⎥ 0 cosϕ ⎥ ⎦
λR = tanh−1 (∓ β1) = ln
0⎤ 0⎥ 0⎥ 1⎥ ⎦
1 ± ν1 ; 0 ⩽ v1 ≤ 1 1 ∓ ν1
(12)
1 ± β1 v1 ; β 1 = ; 0 ⩽ v 1 ⩽ c; 0 ⩽ β 1 ≤ 1 1 ∓ β1 c (13)
The above condition specifies the “spatio-temporal 4D rotation” as a Lorentz transformation. As the matrix notation was already chosen in the previous chapter, in order to exhibit the equivalence of 4D and Lorentz group transformations, the second one will be expressed by a matrix too. Normally, 3D spatial rotation in a matrix notation appears as [14,4]:
(8)
0 0 0 ⎡1 ⎤ cosβ cosγ sinβ − cosβ sinγ ⎢0 ⎥ ⎢ ⎥ sin cos 0 cos sin cos cos α β α γ α γ − ⎢ ⎥ ⎢ ⎥ + sinα sinβ cosγ − sinα sinβ ⎢ ⎥ R3D (α, β, γ ) = ⎢ ⎥ sinγ ⎢ ⎥ ⎢ 0 sinα sinγ cosα cosβ ⎥ sinα cosγ ⎢ ⎥ − cosα sinβ cosγ + cosα sinβ ⎢ ⎥ ⎢ ⎥ sin γ ⎣ ⎦
As emphasised before, a general 4D rotation matrix is:
R 4D (α, β , γ , λ, θ , ϕ) = R (α ) R (β ) R (γ ) R (λ ) R (θ) R (ϕ)
(11)
The whole procedure has a unitary speed (c = 1). Velocity vis expressed in unspecified geometrical units. If we demand the velocity is expressed in units of c(speed of light in a vacuum), a well-known condition is obtained:
(7)
⎡ cosϕ 0 R (ϕ) = ⎢ ⎢ 0 ⎢ ∓ isinϕ ⎣
− ν1 = ∓ iν1 ∓i
and it’s clear that angle λ must be imaginary: λ = λR i , where λR is a real number. Therefore tanλR i = ∓ iν1, or equivalently itanhλR = ∓ iν1. This finally gives the condition:
1. a11 a21 = −a12 a22 2 2 2 2 = 1, a22 = 1, a21 = −1, a12 = −1 2. a11
⎡ cosλ ∓ isinλ R (λ ) = ⎢ ∓ isinλ cosλ ⎢ 0 0 ⎢ 0 0 ⎣
(10)
(5)
(9)
As presenting this matrix in the article in a neat form exceeds my abilities, it is convenient to list all the matrix elements separately (rμν are the elements of the matrix R 4D (α, β, γ , λ, θ, ϕ) ) as follows:
(14) And a Lorentz boost matrix representation (please note that here 1 gamma with index is meant to be γ i = rather than an angle) i 2 1 − (ν )
2
Results in Physics 13 (2019) 102141
M. Myszkowski
Table 3 Differences between a common 4D Euclidean rotation matrix and a 4D rotation matrix. 4D Euclidean rotation matrix
4D spatio-temporal rotation matrix
6 independent parameters: each represents the angle of rotation about a different but equally important axis A rotation always preserves four-vector norm, defined by a metric tensor with Tr[1,1,1,1]
6 independent parameters: 3 represent the angles of rotations in 3D space and 3 represent spatiotemporal rotations. Spatio-temporal and spatial rotations are not equal. A rotation always preserves a four-vector norm, defined by a spacetime interval, i.e. by a metric tensor with Tr[1,−1,−1,−1]
[1,3,5] is:
W=
∫t
t2
1
1 2 3 − γ1v1 − γ1γ 2v 2 − γ1γ 2γ 3v 3 ⎤ ⎡ γγγ ⎢− γ1γ 2γ 3v1 γ1 γ1γ 2v1v 2 γ1γ 2γ 3v1v 3 ⎥ ⎥ Λ(ν1)Λ(ν 2)Λ(ν 3) = ⎢ 2 3 2 0 γ2 γ 2γ 3v1v 3 ⎥ ⎢ −γ γ v ⎢ − γ 3v 3 ⎥ 3 0 0 γ ⎣ ⎦
d4xL (ψ, ∂μ ψ)
(20)
The above formula stays consistent under spatio-temporal rotations. Action variation looks as follows:
δW =
(15)
∫t
t2
1
To get a full infinitesimal matrix, Eq. (14) needs to be multiplied by Eq. (15), and then all the quadratic terms ignored [1]. Therefore:
(
d4x ∂μ ⎡ g μνL − ⎢ ⎣
)⎤⎥⎦ x δω
∂L ∂ νψ ∂ (∂μ ψ)
τ
ντ
(21)
Then [5]:
δW 1 2 3 ⎡ 1 −v −v −v ⎤ 1 ⎢ ⎥ v 1 γ β − − x ′μ = Λ νμ x ν , Λ νμ = ⎢ 2 ⎥ γ 1 − α⎥ ⎢− v ⎢− v 3 − β α 1 ⎥ ⎣ ⎦
=
ω vτ
(16)
μ RInfinitezymal
=
δωτμ g τν
⎡ 0 − iλ − iθ − iϕ ⎤ ⎢ iλ 0 − γ − β ⎥ =⎢ ⎥ 0 −α⎥ ⎢ iθ γ ⎢ α 0 ⎥ ⎦ ⎣iϕ β
)
(
)
∫
μτ μν ∂0 (Tsymmetrical x ν − Tsymmetrical x τ ) = ∂0 M μντ = 0
(23)
I obtain that an angular momentum density tensor M μντ is a conserved Noether current, exactly as in a Lorentz transformation case [5,9,1]. 4. Further considerations Although this article does not present a new conservation law, it nevertheless provides a new, fresh view on Lorentz transformations. The above results are interpreted as a rather straightforward proof that an SO(4) symmetry group can be artificially constructed with the spatial part of a matrix in Eq. (9) and Lorentz transformations [1,5]. I think that a more appropriate way of thinking should be based on a consistent 4D complex rotation matrix. As in a 3D Euclidean case, the rotational symmetry (in 4D) was considered and the angular momentum density tensor was obtained as a conserved quantity (also a 4D tensor). The SO(4) group may also be considered as a 4D Euclidean rotation around a fixed point. Nevertheless, a 4D rotation matrix is different from a common 4D Euclidean matrix by imaginary constants. The imaginary constant was introduced in order to preserve the spacetime interval (Table 3). It is important to emphasise the relation between Lorentz boosts, 3D rotations and 4D rotations. As I pointed out in the beginning of paragraph three, the angles λ, θ , ϕ parameterize transformations to another inertial frame, while α, β , γ stands for common 3D rotations. Therefore, the 4D rotations represents both Lorentz boosts and spatial rotations. It is worth stressing that a rotation matrix whose components are shown in Eq. (10) has some interesting physical properties itself. Unlike Lorentz boosts, where three continuous parameters (velocities) can take a value from − c to c [12,11], spatio-temporal rotations are parameterised with continuous, unrestricted angles. However, due to sin & cos periodicity, the whole math is contained in α, β , γ , θ , ϕ , λ ∈ (0, 2π ) . It was emphasised earlier in Eqs. (10)–(12). Another perfect example is the ω matrix from Eq. (18), that has the same properties as the Lorentz transformation case [5,1].
(17)
(18)
Where:
δω μν
(
By imposing that the variation of the action needs to be zeroed:
Now it is clear that the previous suspicions were right, i.e. v1 ∼ λ, v 2 ∼ θ , v 3 ∼ ϕ . One can be concerned about an imaginary constant, which cannot be factored out of a 4D matrix. Fortunately, it will turn out later that this does not affect the main calculations connected with Noether’s theorem and the conclusion about the conservation of an angular momentum density tensor is true. Finally, using the previously developed machinery, Noether’s theorem can be applied. The first step is to research which forms of the Lagrangian dependent on field ψ stay unchanged under a 4D rotation. There are two types of objects that stay unchanged under the above transformation: terms built with ∑μ = 0,1,2,3 g μμ (∂μ ψ)2 and ψ . Owing to this, it turns out that every free field must be invariable [1,3,9,5]. Most general Lagrangian forms will be the functions of invariable arguments. This extensive group contains, for example, every free field and self-interacting fields. In order to simplify the subsequent calculations, the infinitesimal transformation of the coordinates is written as follows:
x ′μ = x μ + δω μν x ν
1 ∂L ∂L d4x ∂μ ⎡ g μνL − ∂ (∂ ψ) ∂ νψ x τ δωντ − ∂μ g μτ L − ∂ (∂ ψ) ∂ τ ψ ⎤ x v δ μ μ ⎢ ⎥ ⎣ ⎦2 t2 1 μτ μν d4x ∂μ (Tsymmetrical x ν − Tsymmetrical x τ ) δω ντ == (22) 2 t1
t2
1
Gathering the facts together, it is rather noticeable that the λ, θ , ϕ angles should be equal or at least proportional to particular velocities v1, v 2, v 3 . It can be suspected that the rotation of the x 0 − x k surface is equivalent to a Lorentz boost along the x k axis (The following introduction was introduced: x 0 = t , x 1 = x , x 2 = y, x 3 = z , where every Greek index takes a value 0, 1, 2, 3, while Latin indexes take a value: 1, 2, 3), but to confirm this, I need to obtain the infinitesimal 4D rotation matrix and compare it with the result in Eq. (16). Hence:
⎡ 1 ∓ iλ ∓ iθ ∓ iϕ ⎤ ⎢ ∓ iλ 1 − γ − β ⎥ =⎢ ⎥ 1 −α⎥ ⎢ ∓ iθ γ ⎢ α 1 ⎥ ⎦ ⎣ ∓ iϕ β
∫t
References (19)
[1] Lancaster T, Blundell SJ. Quantum field theory for gifted amateur. Oxford University Press; 2014. [2] Hanc J, Tuleja S, Hancova M. Symmetries and conservation laws: consequences of Noether’s theorem. Am J Phys 2004:1–8.
Now, let’s write down the action—an integral over spacetime with Lagrangian density [1,3,5,9]: 3
Results in Physics 13 (2019) 102141
M. Myszkowski [3] Peskin ME, Schroeder DV. An introduction to quantum field theory. Perseus Books Publishing; 1995. [4] Feynman RP, Leighton RB, Sands M. The Feynman lectures on physics. vol. 3. Basic Books; 1963. New Millenium Edition. [5] Kraśkiewicz J. Elementy Klasycznej i Kwantowej Teorii Pola [Elements of Classical and Quantum Field Theory]. UMCS 2003. [6] Gorni G, Zampieri G. Revisiting Noether’s theorem on constants of motion. J Nonlinear Math Phys 2014;21:4–30. [7] Jones HF. Groups, representations and physics. 2nd ed. Institute of Physics Publishing; 2001. [8] van Beveren E. Some notes on group theory. P-3000 COIMBRA; 2012. [9] Weinberg S. The quantum theory of fields. Cambridge University Press; 1995.
[10] Schutz BFA. First course in general relativity. Cambridge University Press; 1985. [11] Feynman RP, Leinghton RB, Sands M. The Feynman lectures on physics. vol. 1. Basic Books; 1963. New Millenium edition. [12] Katz R. An introduction to the special theory of relativity. College Physics; 1964. [13] Gravity Hartle JB. An introduction to Einstein’s general relativity. 1st ed. Pearson Education; 2003. [14] Zhelezov OI. N-dimensional rotation matrix generation algorithm. Am J Comput Appl Math 2017;7:1–7. [15] Chaichian M, Hagedorn R. Symmetries in quantum mechanics: from angular momentum to supersymmetry. Institute of Physics Publishing; 1998. [16] Shankar R. Principles of quantum mechanics. Kluwer Academic Publishers; 1980.
4
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
A new perspective on spacetime 4D rotations and the SO(4) transformation group
T
Mikołaj Myszkowski Borówiec Pod Borem 25, Poznań 62-023, Poland
A R T I C LE I N FO
A B S T R A C T
Keywords: 4D rotations Lorentz transformation Noether’s theorem
In quantum field theory, every conservation law is enforced by adequate symmetries. The Lorentz group, which represents two continuous symmetries: rotations in 3D Euclidean space and Lorentz boosts, generates conservation of the angular momentum tensor. This article shows that both Lorentz boosts and 3D rotations can be replaced by spatio-temporal rotations in Minkowski spacetime. Noether’s theorem certifies that the consequences of 4D rotations are identical to the Lorentz transformation group SO(4). I demonstrate how to build a spatio-temporal rotation matrix that preserves the spacetime interval, and that the resulting matrix represents the special orthogonal group of order 4 (SO(4)).
1. Introduction Symmetries play the main role in modern physics [1–3]. As long as the world was described by classical theories (non-quantum ones), miscellaneous laws of physics were added ad hoc, as a theory paradigm. When quantum mechanics was finally formulated, Noether’s theorem indicated that every symmetry forms a conservation law [1,4–6]. One of the oldest and easiest symmetries are 3D rotations. If the Lagrangian of a system L preserves its form under that transformation, it results in the emergence of conserved Noether currents and Noether charges, i.e. a 3D angular momentum tensor [5]. Transformations of Lorentz group SO(4) [7,8] (a special orthogonal group of order 4, also considered as a 4D Euclidean rotational group) can be represented in a 4 × 4 matrix form, where spatial components are common rotations and temporal components are Lorentz transformations [9,1,5]. In this article I will show that new 4D spacetime rotations express a Lorentz transformation. In order to preserve a spacetime interval (in the 3 dimensional case, rotations must preserve a vector norm, in a non-Euclidean 4D space rotations must preserve a four-vector norm, determined by a metric tensor), a complex analysis must be involved. In the subsequent calculations it is assumed that a metric tensor in flat spacetime is:
0 0 ⎤ ⎡1 0 0 −1 0 0 ⎥ gμν = ⎢ ⎢0 0 − 1 0 ⎥ 0 − 1⎦ ⎣0 0
(1)
Lorentz boosts, unlike 3D rotations, influence time as well [10–13]. This implies the association with time. These two transformations assembled together can be displayed as the result of multiplications.
Because Lorentz boosts and 3D rotations preserve a spacetime interval, the result of a complete transformation needs to preserve this quantity too. 2. Spatio-temporal (4D) rotation matrix To construct a full 4D Euclidean rotation matrix, a minimum of six 4 × 4 matrixes [14] (each represents a different rotation) need to be multiplied. It is worth mentiontioning that there are already some similarities with a SO(4) Lorentz group (Table 1): It is convenient to assign different angles to adequate rotations (Table 2): Matrices R (α ), R (β ), R (γ ) are the same as in a SO(3) rotation group case [15,16], hence:
⎡1 0 R (α ) = ⎢ ⎢0 ⎢ ⎣0
0 0 0 ⎤ 1 0 0 ⎥ , 0 cosα − sinα ⎥ ⎥ 0 sinα cosα ⎦
0 ⎡1 0 0 cosγ − sinγ R (γ ) = ⎢ ⎢ 0 sinγ cosγ ⎢ 0 ⎣0 0
0 ⎡1 ⎢ 0 cosβ R (β ) = ⎢ 0 0 ⎢ 0 − sinβ ⎣
0⎤ 0⎥ 0⎥ ⎥ 1⎦
0 0 ⎤ 0 sinβ ⎥ 1 0 ⎥ 0 cosβ ⎥ ⎦
(2)
(3)
However, the rest of the matrices take a less obvious form. Let’s assume that all components of R (λ ), R (θ), R (ϕ) are different from R (α ), R (β ), R (γ ) only by a multiplicative constant (as it will emerge later, this is a complex constant):
E-mail address: [email protected]. https://doi.org/10.1016/j.rinp.2019.02.077 Received 3 September 2018; Received in revised form 12 February 2019; Accepted 23 February 2019 Available online 26 February 2019 2211-3797/ © 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Results in Physics 13 (2019) 102141
M. Myszkowski
r00 = cosϕcosθcosλ r01 = ∓ icosϕcosθsinλ r02 = ∓ isinθcosϕ r03 = ∓ isinϕ r10 = ± icosβ (sinγ sinθcosλ − cosγ sinλ ) ∓ isinβ sinϕcosθ cosλ r11 = cosβ (cosγ cosλ + sinγ sinθsinλ ) − sinβ sinϕcosθsinλ r12 = −sinβ sinθsinϕ − cosβ sinγ cosθ r13 = sinβ cosϕ r20 = ∓ icosα (sinγ sinλ + cosγ sinθcosλ ) ∓ isinα (−sinβ (sinγ sinθcosλ − cosγ sinλ ) − cosβ sinϕcosθ cosλ ) r21 = cosα (sinγ cosλ − cosγ sinθsinλ )− − sinα (−sinβ (cosγ cosλ + sinγ sinθsinλ ) − cosβ sinϕcosθsinλ ) r22 = cosα cosγ cosθ + sinα ( −sinβ sinγ cosθ + cosβsinθsinϕ) r23 = −sinα cosβ cosϕ r30 = ∓ isinα (sinγ sinλ + cosγ sinθcosλ ) ± icosα ( −sinβ (sinγ sinθ cosλ − cosγ sinλ ) − cosβ sinϕcosθcosλ ) r31 = sinα (sinγ cosλ − cosγ sinθsinλ )− + cosα ( −sinβ (cosγ cosλ + sinγ sinθsinλ ) − cosβ sinϕcosθsinλ ) r32 = sinα cosγ cosθ − cosα (−sinβ sinγ cosθ + cosβ sinθsinϕ) r33 = cosα cosβ cosϕ
Table 1 Similarities of SO(4) group and 4D rotations. SO(4) group transformations
Spatio-temporal 4D rotations
6 independent parameters: 3 velocity components and 3 angles
6 independent parameters: 3 spatial angles and 3 “spatio-temporal” angles
Table 2 Adequate rotations and assigned angles. α t-x
Angle: Rotation plane
β t-y
⎡ a11cosλ − a12sinλ a21sinλ a22cosλ R (λ ) = ⎢ ⎢ 0 0 ⎢ 0 ⎣ 0
0 0 1 0
0⎤ 0 ⎥, 0⎥ ⎥ 1⎦
γ t-z
λ y-z
ϕ x-y
θ x-z
⎡ a11cosθ 0 R (θ) = ⎢ ⎢ a21sinθ ⎢ ⎣ 0
0 − a12sinθ 1 0 0 a22cosθ 0 0
0⎤ 0⎥ 0⎥ ⎥ 1⎦ (4)
⎡ a11cosϕ 0 R (ϕ) = ⎢ ⎢ 0 ⎢ a sinϕ ⎣ 21
0 1 0 0
0 − a12sinϕ ⎤ 0 0 ⎥ ⎥ 1 0 0 a22cosϕ ⎥ ⎦
3. Similarities of 4D rotations and Lorentz boosts
Using a norm invariance condition for the R (λ ) transformation [11,12], (ds′2 = ds 2 ) I obtain (please note that the matrix composed of t , x , y, z is represented by a row vector):
In order to find a condition which makes a spatio-temporal rotation identical to a Lorentz boost, one should demand R (λ ) = Λ(ν1) . It emerges that cosλ = γ1 and ∓ isinλ = −γ1ν1. Thus:
(ds′)2 = (t ′)2 − (x ′)2 = (a11 t cosλ + a21 x sinλ )2 − (−a12 t sinλ + a22 x cosλ )2 2 2 2 2 2 2 = a11 t cos2 λ + 2a11 a21 xt cosλsinλ + a21 x sin2 λ − a12 t sin2 λ 2 2 + 2a12 a22 xt cosλ sinλ − a22 x cos2 λ = t 2 − x 2
tanλ =
(6)
Therefore, multiplicative components need to comply with the undermentioned conditions:
λR = tanh−1 (∓ ν1) = ln
In order to decrease the number of possibilities, the conditions R (λ = 0) = , R (θ = 0) = , R (ϕ = 0) = are imposed, ( is identity matrix) which give a11 = a22 = 1. After summing up, there is still a minuscule freedom of choice (because a11 = a22 = 1, a12 = −a21 = ± i ). Because this freedom should not affect further considerations, the following representation can be chosen:
0 0 1 0
0⎤ 0 ⎥, 0⎥ 1⎥ ⎦
⎡ cosθ 0 R (θ) = ⎢ ⎢ ∓ isinθ ⎢ 0 ⎣
0 ∓ isinθ 1 0 0 cosθ 0 0
0 1 0 0
0 ∓ isinϕ ⎤ 0 0 ⎥ 1 0 ⎥ 0 cosϕ ⎥ ⎦
λR = tanh−1 (∓ β1) = ln
0⎤ 0⎥ 0⎥ 1⎥ ⎦
1 ± ν1 ; 0 ⩽ v1 ≤ 1 1 ∓ ν1
(12)
1 ± β1 v1 ; β 1 = ; 0 ⩽ v 1 ⩽ c; 0 ⩽ β 1 ≤ 1 1 ∓ β1 c (13)
The above condition specifies the “spatio-temporal 4D rotation” as a Lorentz transformation. As the matrix notation was already chosen in the previous chapter, in order to exhibit the equivalence of 4D and Lorentz group transformations, the second one will be expressed by a matrix too. Normally, 3D spatial rotation in a matrix notation appears as [14,4]:
(8)
0 0 0 ⎡1 ⎤ cosβ cosγ sinβ − cosβ sinγ ⎢0 ⎥ ⎢ ⎥ sin cos 0 cos sin cos cos α β α γ α γ − ⎢ ⎥ ⎢ ⎥ + sinα sinβ cosγ − sinα sinβ ⎢ ⎥ R3D (α, β, γ ) = ⎢ ⎥ sinγ ⎢ ⎥ ⎢ 0 sinα sinγ cosα cosβ ⎥ sinα cosγ ⎢ ⎥ − cosα sinβ cosγ + cosα sinβ ⎢ ⎥ ⎢ ⎥ sin γ ⎣ ⎦
As emphasised before, a general 4D rotation matrix is:
R 4D (α, β , γ , λ, θ , ϕ) = R (α ) R (β ) R (γ ) R (λ ) R (θ) R (ϕ)
(11)
The whole procedure has a unitary speed (c = 1). Velocity vis expressed in unspecified geometrical units. If we demand the velocity is expressed in units of c(speed of light in a vacuum), a well-known condition is obtained:
(7)
⎡ cosϕ 0 R (ϕ) = ⎢ ⎢ 0 ⎢ ∓ isinϕ ⎣
− ν1 = ∓ iν1 ∓i
and it’s clear that angle λ must be imaginary: λ = λR i , where λR is a real number. Therefore tanλR i = ∓ iν1, or equivalently itanhλR = ∓ iν1. This finally gives the condition:
1. a11 a21 = −a12 a22 2 2 2 2 = 1, a22 = 1, a21 = −1, a12 = −1 2. a11
⎡ cosλ ∓ isinλ R (λ ) = ⎢ ∓ isinλ cosλ ⎢ 0 0 ⎢ 0 0 ⎣
(10)
(5)
(9)
As presenting this matrix in the article in a neat form exceeds my abilities, it is convenient to list all the matrix elements separately (rμν are the elements of the matrix R 4D (α, β, γ , λ, θ, ϕ) ) as follows:
(14) And a Lorentz boost matrix representation (please note that here 1 gamma with index is meant to be γ i = rather than an angle) i 2 1 − (ν )
2
Results in Physics 13 (2019) 102141
M. Myszkowski
Table 3 Differences between a common 4D Euclidean rotation matrix and a 4D rotation matrix. 4D Euclidean rotation matrix
4D spatio-temporal rotation matrix
6 independent parameters: each represents the angle of rotation about a different but equally important axis A rotation always preserves four-vector norm, defined by a metric tensor with Tr[1,1,1,1]
6 independent parameters: 3 represent the angles of rotations in 3D space and 3 represent spatiotemporal rotations. Spatio-temporal and spatial rotations are not equal. A rotation always preserves a four-vector norm, defined by a spacetime interval, i.e. by a metric tensor with Tr[1,−1,−1,−1]
[1,3,5] is:
W=
∫t
t2
1
1 2 3 − γ1v1 − γ1γ 2v 2 − γ1γ 2γ 3v 3 ⎤ ⎡ γγγ ⎢− γ1γ 2γ 3v1 γ1 γ1γ 2v1v 2 γ1γ 2γ 3v1v 3 ⎥ ⎥ Λ(ν1)Λ(ν 2)Λ(ν 3) = ⎢ 2 3 2 0 γ2 γ 2γ 3v1v 3 ⎥ ⎢ −γ γ v ⎢ − γ 3v 3 ⎥ 3 0 0 γ ⎣ ⎦
d4xL (ψ, ∂μ ψ)
(20)
The above formula stays consistent under spatio-temporal rotations. Action variation looks as follows:
δW =
(15)
∫t
t2
1
To get a full infinitesimal matrix, Eq. (14) needs to be multiplied by Eq. (15), and then all the quadratic terms ignored [1]. Therefore:
(
d4x ∂μ ⎡ g μνL − ⎢ ⎣
)⎤⎥⎦ x δω
∂L ∂ νψ ∂ (∂μ ψ)
τ
ντ
(21)
Then [5]:
δW 1 2 3 ⎡ 1 −v −v −v ⎤ 1 ⎢ ⎥ v 1 γ β − − x ′μ = Λ νμ x ν , Λ νμ = ⎢ 2 ⎥ γ 1 − α⎥ ⎢− v ⎢− v 3 − β α 1 ⎥ ⎣ ⎦
=
ω vτ
(16)
μ RInfinitezymal
=
δωτμ g τν
⎡ 0 − iλ − iθ − iϕ ⎤ ⎢ iλ 0 − γ − β ⎥ =⎢ ⎥ 0 −α⎥ ⎢ iθ γ ⎢ α 0 ⎥ ⎦ ⎣iϕ β
)
(
)
∫
μτ μν ∂0 (Tsymmetrical x ν − Tsymmetrical x τ ) = ∂0 M μντ = 0
(23)
I obtain that an angular momentum density tensor M μντ is a conserved Noether current, exactly as in a Lorentz transformation case [5,9,1]. 4. Further considerations Although this article does not present a new conservation law, it nevertheless provides a new, fresh view on Lorentz transformations. The above results are interpreted as a rather straightforward proof that an SO(4) symmetry group can be artificially constructed with the spatial part of a matrix in Eq. (9) and Lorentz transformations [1,5]. I think that a more appropriate way of thinking should be based on a consistent 4D complex rotation matrix. As in a 3D Euclidean case, the rotational symmetry (in 4D) was considered and the angular momentum density tensor was obtained as a conserved quantity (also a 4D tensor). The SO(4) group may also be considered as a 4D Euclidean rotation around a fixed point. Nevertheless, a 4D rotation matrix is different from a common 4D Euclidean matrix by imaginary constants. The imaginary constant was introduced in order to preserve the spacetime interval (Table 3). It is important to emphasise the relation between Lorentz boosts, 3D rotations and 4D rotations. As I pointed out in the beginning of paragraph three, the angles λ, θ , ϕ parameterize transformations to another inertial frame, while α, β , γ stands for common 3D rotations. Therefore, the 4D rotations represents both Lorentz boosts and spatial rotations. It is worth stressing that a rotation matrix whose components are shown in Eq. (10) has some interesting physical properties itself. Unlike Lorentz boosts, where three continuous parameters (velocities) can take a value from − c to c [12,11], spatio-temporal rotations are parameterised with continuous, unrestricted angles. However, due to sin & cos periodicity, the whole math is contained in α, β , γ , θ , ϕ , λ ∈ (0, 2π ) . It was emphasised earlier in Eqs. (10)–(12). Another perfect example is the ω matrix from Eq. (18), that has the same properties as the Lorentz transformation case [5,1].
(17)
(18)
Where:
δω μν
(
By imposing that the variation of the action needs to be zeroed:
Now it is clear that the previous suspicions were right, i.e. v1 ∼ λ, v 2 ∼ θ , v 3 ∼ ϕ . One can be concerned about an imaginary constant, which cannot be factored out of a 4D matrix. Fortunately, it will turn out later that this does not affect the main calculations connected with Noether’s theorem and the conclusion about the conservation of an angular momentum density tensor is true. Finally, using the previously developed machinery, Noether’s theorem can be applied. The first step is to research which forms of the Lagrangian dependent on field ψ stay unchanged under a 4D rotation. There are two types of objects that stay unchanged under the above transformation: terms built with ∑μ = 0,1,2,3 g μμ (∂μ ψ)2 and ψ . Owing to this, it turns out that every free field must be invariable [1,3,9,5]. Most general Lagrangian forms will be the functions of invariable arguments. This extensive group contains, for example, every free field and self-interacting fields. In order to simplify the subsequent calculations, the infinitesimal transformation of the coordinates is written as follows:
x ′μ = x μ + δω μν x ν
1 ∂L ∂L d4x ∂μ ⎡ g μνL − ∂ (∂ ψ) ∂ νψ x τ δωντ − ∂μ g μτ L − ∂ (∂ ψ) ∂ τ ψ ⎤ x v δ μ μ ⎢ ⎥ ⎣ ⎦2 t2 1 μτ μν d4x ∂μ (Tsymmetrical x ν − Tsymmetrical x τ ) δω ντ == (22) 2 t1
t2
1
Gathering the facts together, it is rather noticeable that the λ, θ , ϕ angles should be equal or at least proportional to particular velocities v1, v 2, v 3 . It can be suspected that the rotation of the x 0 − x k surface is equivalent to a Lorentz boost along the x k axis (The following introduction was introduced: x 0 = t , x 1 = x , x 2 = y, x 3 = z , where every Greek index takes a value 0, 1, 2, 3, while Latin indexes take a value: 1, 2, 3), but to confirm this, I need to obtain the infinitesimal 4D rotation matrix and compare it with the result in Eq. (16). Hence:
⎡ 1 ∓ iλ ∓ iθ ∓ iϕ ⎤ ⎢ ∓ iλ 1 − γ − β ⎥ =⎢ ⎥ 1 −α⎥ ⎢ ∓ iθ γ ⎢ α 1 ⎥ ⎦ ⎣ ∓ iϕ β
∫t
References (19)
[1] Lancaster T, Blundell SJ. Quantum field theory for gifted amateur. Oxford University Press; 2014. [2] Hanc J, Tuleja S, Hancova M. Symmetries and conservation laws: consequences of Noether’s theorem. Am J Phys 2004:1–8.
Now, let’s write down the action—an integral over spacetime with Lagrangian density [1,3,5,9]: 3
Results in Physics 13 (2019) 102141
M. Myszkowski [3] Peskin ME, Schroeder DV. An introduction to quantum field theory. Perseus Books Publishing; 1995. [4] Feynman RP, Leighton RB, Sands M. The Feynman lectures on physics. vol. 3. Basic Books; 1963. New Millenium Edition. [5] Kraśkiewicz J. Elementy Klasycznej i Kwantowej Teorii Pola [Elements of Classical and Quantum Field Theory]. UMCS 2003. [6] Gorni G, Zampieri G. Revisiting Noether’s theorem on constants of motion. J Nonlinear Math Phys 2014;21:4–30. [7] Jones HF. Groups, representations and physics. 2nd ed. Institute of Physics Publishing; 2001. [8] van Beveren E. Some notes on group theory. P-3000 COIMBRA; 2012. [9] Weinberg S. The quantum theory of fields. Cambridge University Press; 1995.
[10] Schutz BFA. First course in general relativity. Cambridge University Press; 1985. [11] Feynman RP, Leinghton RB, Sands M. The Feynman lectures on physics. vol. 1. Basic Books; 1963. New Millenium edition. [12] Katz R. An introduction to the special theory of relativity. College Physics; 1964. [13] Gravity Hartle JB. An introduction to Einstein’s general relativity. 1st ed. Pearson Education; 2003. [14] Zhelezov OI. N-dimensional rotation matrix generation algorithm. Am J Comput Appl Math 2017;7:1–7. [15] Chaichian M, Hagedorn R. Symmetries in quantum mechanics: from angular momentum to supersymmetry. Institute of Physics Publishing; 1998. [16] Shankar R. Principles of quantum mechanics. Kluwer Academic Publishers; 1980.
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