Bargmann–Wigner equations

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This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators.

In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations (or BW equations or BWE) are relativistic wave equations which describe free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 ...) or half-integer for fermions (j = 12, 32, 52 ...). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature (see references).

They were proposed by Valentine Bargmann and Eugene Wigner in 1948,[1] using Lorentz group theory,[2] and building on the work of those who pioneered quantum theory within the first half of the twentieth century.[3][4]

Origin from the Dirac equation[edit]

Main article: Dirac equation

For reference, the Dirac equation is summarized below. It is the basis for building relativistic wave equations with wavefunctions of higher spin.

The covariant form of the Dirac equation for an uncharged particle is:[5]


(-\gamma^\mu \hat{P}_\mu + mc)\Psi = 0 \,,

 

 

 

 

(1)

where Ψ = Ψ(r, t) is a rank-1 4-component spinor field, a function of the particle's position r and time t, with components ψα = ψα(r, t) in which α is a bispinor index that takes values 1, 2, 3, 4. Further, γμ = (γ0, γ) are the gamma matrices, and

\hat{P}_\mu = i\hbar \partial_\mu

is the 4-momentum operator. The operator constituting the entire equation, (−γμPμ + mc) = (−γμμ + mc), is a 4 × 4 matrix, because of the γμ matrices, and the mc term scalar-multiplies the 4 × 4 identity matrix (usually not written for simplicity). Explicitly, in the Dirac representation of the gamma matrices:[3]



\begin{align}
-\gamma^\mu \hat{P}_\mu + mc & = -\gamma^0 \frac{\hat{E}}{c} - \boldsymbol{\gamma}\cdot(-\hat{\mathbf{p}}) + mc \\
& = -\begin{pmatrix}
I_2 & 0 \\
0 & -I_2 \\
\end{pmatrix}\frac{\hat{E}}{c} 
+
\begin{pmatrix}
0 & \boldsymbol{\sigma}\cdot\hat{\mathbf{p}} \\
-\boldsymbol{\sigma}\cdot\hat{\mathbf{p}} & 0 \\
\end{pmatrix} + \begin{pmatrix}
I_2 & 0 \\
0 & I_2 \\
\end{pmatrix}mc \\
& =
\begin{pmatrix}
-\hat{E}/c+mc & 0 & \hat{p}_z & \hat{p}_x - i\hat{p}_y \\
0 & -\hat{E}/c+mc & \hat{p}_x + \hat{p}_y & -\hat{p}_z \\
-\hat{p}_z & -(\hat{p}_x - i\hat{p}_y) & \hat{E}/c+mc & 0 \\
-(\hat{p}_x + i\hat{p}_y) & \hat{p}_z & 0 & \hat{E}/c+mc \\
\end{pmatrix}
\end{align}

where σ = (σ1, σ2, σ3) = (σx, σy, σz) is a vector of the Pauli matrices, E is the energy operator, p = (p1, p2, p3) = (px, py, pz) is the 3-momentum operator, I2 denotes the 2 × 2 identity matrix, the zeros (in the second line) are actually 2 × 2 blocks of zero matrices.

The Dirac equation (1) can be written as a coupled set of equations:


(-\hat{E}  + mc )\psi_{1,2} = (-\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})\psi_{3,4}

 

 

 

 

(1A)


(\hat{E} + mc )\psi_{3,4} = (\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})\psi_{1,2}

 

 

 

 

(1B)

where

 \Psi = 
\begin{pmatrix}
\psi_{1,2} \\
\psi_{3,4} \\
\end{pmatrix}\,\quad \psi_{1,2} = 
\begin{pmatrix}
\psi_1 \\
\psi_2 \\
\end{pmatrix}\,\quad \psi_{3,4} = 
\begin{pmatrix}
\psi_3 \\
\psi_4 \\
\end{pmatrix}\,.

One 2-component spinor ψ1,2 describes the spin-1/2 fermion, the other ψ3,4 describes the antifermion.

For a charged particle moving in an electromagnetic field, minimal coupling can be introduced:


[-\gamma^\mu (i\hbar \partial_\mu - eA_\mu)+mc]\Psi = 0

 

 

 

 

(1C)

where e is the electric charge of the particle and Aμ = (A0, A) is the electromagnetic four-potential.

BW equations[edit]

For a free particle of spin j, the BW equations are a set of 2j coupled linear partial differential equations, each with a similar mathematical form to the Dirac equation.

Uncharged massive particles[edit]

For a free particle with zero electric charge, the full set of equations are:[3][4][6]

\begin{align}
& (-\gamma^\mu \hat{P}_\mu + mc)_{\alpha_1 \alpha_1'}\psi_{\alpha'_1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\
& (-\gamma^\mu \hat{P}_\mu + mc)_{\alpha_2 \alpha_2'}\psi_{\alpha_1 \alpha'_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\
& \qquad \vdots \\
& (-\gamma^\mu \hat{P}_\mu + mc)_{\alpha_{2j} \alpha'_{2j}}\psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha'_{2j}} = 0 \\
\end{align}

which follow the pattern;

(-\gamma^\mu \hat{P}_\mu + mc)_{\alpha_r \alpha'_r}\psi_{\alpha_1 \cdots \alpha'_r \cdots \alpha_{2j}} = 0

 

 

 

 

(2)

for r = 1, 2, ... 2j. Again, the operator (−γμPμ + mc) is a 4 × 4 matrix. The wavefunction Ψ = Ψ(r, t) has components

 \psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} (\mathbf{r},t)

and is now a rank-2j 4-component spinor field, usually symmetric in all bispinor indices, but not necessarily; for example, the spin-0 case is antisymmetric. Each index takes the values 1, 2, 3, or 4, so there are 42j components of the entire spinor field Ψ, although a completely symmetric wavefunction reduces the number of independent components to 2(2j + 1).

Some authors (for example Loide and Saar[4]) use n = 2j, where n is a non-negative integer (thereby j is a half-integer or integer), because this helps remove factors of 2.

The above matrix operator contracts with one bispinor index of Ψ at a time (analogous but not equivalent to matrix multiplication), so some properties of the Dirac equation also apply to the BW equations:

E^2 = (pc)^2 + (mc^2)^2

The components for a totally symmetric wavefunction are explicitly:[3]

\Psi = \begin{pmatrix}
\psi_{1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\
\psi_{2 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\
\psi_{3 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\
\psi_{4 \alpha_2 \alpha_3 \cdots \alpha_{2j}} \\
\end{pmatrix}

where the indices are selected so that: α2 ≤ α3 ≤ ... α2j.

Unlike the Dirac equation, which can incorporate the electromagnetic field via minimal coupling (1C), the B–W formalism comprises intrinsic contradictions and difficulties when the electromagnetic field interaction is incorporated. In other words, it is not possible to make the change PμPμeAμ.[7][8] An indirect approach to investigate electromagnetic influences of the particle is to derive the electromagnetic four-currents currents and multipole moments for the particle, rather than include the interactions in the wave equations themselves.[9][10]

Coupled equations[edit]

Analogous to (1A) and (1B), the BW equations can be written as a set of coupled equations:

(-\hat{E} + mc )^{2j} \psi_{1,2}^{[2j]} = (-\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^{[2j]} \psi_{3,4}^{[2j]}

 

 

 

 

(2A)

(\hat{E} + mc )^{2j} \psi_{3,4}^{[2j]} = (\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^{[2j]} \psi_{1,2}^{[2j]}

 

 

 

 

(2B)

where the notation [2j] denotes the 2j induced spinor or matrix (defined in the next section). Each of ψ1,2 and ψ3,4 has 2j + 1 independent components.

These can be recombined:[3]

[(E^2 - (mc^2)^2)^{2j} - (\hat{\mathbf{p}}^2)^{2j}]\begin{pmatrix}
\psi_{1,2}^{[2j]} \\
\psi_{3,4}^{[2j]}
\end{pmatrix}
 = 0

 

 

 

 

(2C)

which upon expanding by the binomial theorem, then factorizing;

[(E^2 - (mc^2)^2) - (\hat{\mathbf{p}}^2)]\left( (E^2 - (mc^2)^2)^{2j-1} + (E^2 - (mc^2)^2)^{2j-2}\hat{\mathbf{p}}^2 + \cdots (\hat{\mathbf{p}}^2)^{2j-1} \right)\begin{pmatrix}
\psi_{1,2}^{[2j]} \\
\psi_{3,4}^{[2j]}
\end{pmatrix}
 = 0

 

 

 

 

(2C)

shows that each component of the BW wavefunction also satisfies the Klein–Gordon equation, uniquely. Conversely, the solutions to the Klein–Gordon equation satisfy the BW equations but are not unique.

Modified gamma matrices[edit]

If we define the following Kronecker product (denoted ⊗) of 4 × 4 identity matrices (denoted I4), with the γμ matrix in the rth place of the product,[4]

 \gamma_r^\mu = \underbrace{I_4 \otimes I_4 \otimes \cdots}_{r-1\,\text{matrices}} \gamma^\mu \cdots \otimes  I_4

for r = 1, 2 ... 2j, these equations (2) can also be written:


(\gamma_r^\mu \hat{P}_\mu - mc )\Psi =0

 

 

 

 

(3)

The γrμ matrices have dimension 42j × 42j. The equations are linear, so adding (3) with respect to the r values gives:


\left(\frac{1}{2j}\sum_{r=1}^{2j}\gamma_r^\mu \hat{P}_\mu - mc \right)\Psi =0

 

 

 

 

(3A)

where the factor of 1/2j is inserted because the matrix elements ±1, ±i are added 2j times. Subtracting (3), one r from the next r + 1; the wavefunction satisfies:


(\gamma_r^\mu - \gamma_{r+1}^\mu)\hat{P}_\mu\psi=0

 

 

 

 

(3B)

for r = 1, 2 ... 2j − 1.

Joos-Weinberg equation[edit]

Introducing a 2(2j + 1) × 2(2j + 1) matrix;[11]

 \gamma^{\mu_1 \mu_2 \cdots \mu_{2j}}

symmetric in any two tensor indices, which generalizes the gamma matrices in the Dirac equation,[3][12] the BW equation takes the form:[13][14]

[(i\hbar)^{2j}\gamma^{\mu_1 \mu_2 \cdots \mu_{2j}} \partial_{\mu_1}\partial_{\mu_2}\cdots\partial_{\mu_{2j}} + (mc)^{2j}]\Psi = 0

or

[\gamma^{\mu_1 \mu_2 \cdots \mu_{2j}} P_{\mu_1}P_{\mu_2}\cdots P_{\mu_{2j}} + (mc)^{2j}]\Psi = 0

 

 

 

 

(4)

This is also known as the Joos-Weinberg equation (or JW or JWE), after H. Joos and Steven Weinberg, found in the early 1960s.[3][11]

Induced matrices[edit]

Definition[edit]

The induced matrices[3] arise from the spinor transformation:


\begin{pmatrix}
a & c \\
b & d \\
\end{pmatrix}\begin{pmatrix}
\psi_1 \\
\psi_2 \\
\end{pmatrix}=\begin{pmatrix}
\chi_1 \\
\chi_2 \\
\end{pmatrix}

 

 

 

 

(5)

that is:

\begin{align}
 a\psi_1 + c\psi_2 &= \chi_1,
\\
 b\psi_1 + d\psi_2 &= \chi_2.
\end{align}

The 2j induced matrix arises by expanding:

(a\psi_1 + c\psi_2)^{j+m}(b\psi_1 + d\psi_2)^{j-m} = \chi_1^{j+m}\chi_2^{j-m}\,,

for m = −j, −j + 1, ... j − 1, j, simplifying, then writing the set of equations in matrix form.

Properties[edit]

Two reasons for introducing the induced matrices is the simple correspondence between induced matrices and powers of eigenvalues, and ease of diagonalization.

Eigenvalues

If A is a 2 × 2 matrix, the 2j induced matrix A[2j] has eigenvalues λ1j + mλ2jm for the same m values as above.

Diagonalization

If the transformation AB−1AB holds, then B[2j] will diagonalize A[2j].

Use in the BW formalism[edit]

In the above equations (1A), (1B), (2A), (2B):


(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^{[2j]} = (i |\hat{\mathbf{p}}|)^{2j}e^{-i\pi\mathbf{J}^{(j)}\cdot\mathbf{n}}

 

 

 

 

(6)

where matrix indices on the left side are understood to be m, m′ = −j, −j + 1 ... j. The mm′ element of the (2j + 1) × (2j + 1) matrix contains the energy–momentum operators and are given by:


{(\boldsymbol{\sigma}\cdot\hat{\mathbf{p}})^{[2j]}_{mm'} = (-1)^{m'-m}\sum_{r=-\infty}^\infty\frac{(-1)^rp_{-}^j(-\hat{p}_z)^{j-m'-r}\hat{p}_z^{j+m-r}(-p_{+})^{m'-m+r}}{r!(j-m'-r)!(j+m-r)!(m'-m+r)!}\sqrt{(j+m)!(j-m)!(j+m')!(j-m')!}}

 

 

 

 

(7)

where n = p/|p| is a unit vector and J(j) = (J(j)1, J(j)2, J(j)3) is the vector of the Pauli matrices for spin s.[15]

The matrix (σ • p)[2j] has eigenvalues ±|p|2j. The degeneracy of the eigenvalues are as follows:

+|p|[2j] −|p|[2j]
Integer spin (j + 1)-fold j-fold
Half-integer spin (j + ½)-fold (j + ½)-fold

Lorentz group structure[edit]

Under a proper orthochronous Lorentz transformation x → Λx in Minkowski space, all one-particle quantum states ψjσ of spin j with spin z-component σ locally transform under some representation D of the Lorentz group:[11][16]

\psi(x) \rightarrow D(\Lambda) \psi(\Lambda^{-1}x)

where D(Λ) is some finite-dimensional representation, i.e. a matrix. Here ψ is thought of as a column vector containing components with the allowed values of σ. The quantum numbers j and σ as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of σ may occur more than once depending on the representation. Representations with several possible values for j are considered below.

The irreducible representations are labeled by a pair of half-integers or integers (A, B). From these all other representations can be built up using a variety of standard methods, like taking tensor products and direct sums. In particular, space-time itself constitutes a 4-vector representation (1/2, 1/2) so that Λ ∈ D'(1/2, 1/2). To put this into context; Dirac spinors transform under the (1/2, 0) ⊕ (0, 1/2) representation. In general, the (A, B) representation space has subspaces that under the subgroup of spatial rotations, SO(3), transform irreducibly like objects of spin j, where each allowed value:

j = A + B, A + B - 1, ..., |A - B|,

occurs exactly once.[17] In general, tensor products of irreducible representations are reducible; they decompose as direct sums of irreducible representations.

The representation for the BW equations is the choice:[7]

D^\mathrm{BW} = \bigotimes_{r=1}^{2j} \left[ D_r^{(1/2,0)}\oplus D_r^{(0,1/2)}\right]\,.

where each Dr is an irreducible representation. This representation does not have definite spin unless j equals 1/2 or 0. One may perform a Clebsch–Gordan decomposition to find the irreducible (A, B) terms and hence the spin content. This redundancy necessitates that a particle of definite spin j that transforms under the DBW representation satisfies field equations.

For the JW equations the choice is:[7]

D^\mathrm{JW} = D^{(j,0)}\oplus D^{(0,j)}\,.

This representation has definite spin j. It turns out that a spin j particle in this representation satisfy field equations too. These equations are very much like the Dirac equations. It is suitable when the symmetries of charge conjugation, time reversal symmetry, and parity are good.

The representations D(j, 0) and D(0, j) can each separately represent particles of spin j. A state or quantum field in such a representation would satisfy no field equation except the Klein-Gordon equation.

Lagrangian[edit]

The Lagrangian which generates equations (2) through the Euler–Lagrange equation (for fields) is not easily found. Methods have been introduced by Guralnik and Kibble, and Larsen and Repko.[18]

One method proposed by Kamefuchi and Takahashi in 1966 was to expand the wavefunctions in terms of 4 × 4 matrices with a required symmetry (conserved properties of the quantum system), then substitute back into the BW equations to yield field equations with that symmetry. From then a Lagrangian can be found by working backwards from the Euler–Lagrange field equations.

D.S. Kaparulin, S.L. Lyakhovich, and A.A. Sharapov take this fundamental approach by starting from symmetries directly, by means of a Poincaré invariant Lagrange anchor.[19] A Lagrange anchor geometrically defines a mapping between fiber bundles, comprising vector bundles, tangent bundles, and the configuration space for the quantum fields. This is less restrictive than a variational formulation (based on the principle of least action) to obtain the equations for the quantum fields.

Formulation in curved spacetime[edit]

Following M. Kenmoku,[16] in local Minkowski space, the gamma matrices satisfy the anticommutation relations:

[\gamma^i,\gamma^j]_{+} = 2\eta^{ij}

where ηij = diag(−1, 1, 1, 1) is the Minkowski metric. For the Latin indices here, i, j = 1, 2, 3. In curved spacetime they are similar:

[\gamma^\mu,\gamma^\nu]_{+} = 2g^{\mu\nu}

where the spatial gamma matrices are contracted with the vierbein biμ to obtain γμ = biμ γi, and gμν = biμbiν is the metric tensor. For the Greek indices; μ, ν = 0, 1, 2, 3.

A covariant derivative for spinors is given by

\mathcal{D}_\mu=\partial_\mu+\Omega_\mu

with the connection Ω given in terms of the spin connection ω by:

\Omega_\mu =\frac{1}{4}\partial_\mu\omega^{ij} (\gamma_i\gamma_j-\gamma_j\gamma_i)

The covariant derivative transforms like ψ:

\mathcal{D}_\mu\psi \rightarrow D(\Lambda) \mathcal{D}_\mu \psi

With this setup, equation (2) becomes:

\begin{align}
& (-i\hbar\gamma^\mu \mathcal{D}_\mu + mc)_{\alpha_1 \alpha_1'}\psi_{\alpha'_1 \alpha_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\
& (-i\hbar\gamma^\mu \mathcal{D}_\mu + mc)_{\alpha_2 \alpha_2'}\psi_{\alpha_1 \alpha'_2 \alpha_3 \cdots \alpha_{2j}} = 0 \\
& \qquad \vdots \\
& (-i\hbar\gamma^\mu \mathcal{D}_\mu + mc)_{\alpha_{2j} \alpha'_{2j}}\psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha'_{2j}} = 0 \,.\\
\end{align}

See also[edit]

References[edit]

Notes[edit]

  1. ^ Bargmann, V.; Wigner, E. P. (1948). "Group theoretical discussion of relativistic wave equations". Proceedings of the National Academy of Sciences of the United States of America 34 (5): 211–23. Bibcode:1948PNAS...34..211B. doi:10.1073/pnas.34.5.211. 
  2. ^ E. Wigner (1937). "On Unitary Representations Of The Inhomogeneous Lorentz Group". Annals of Mathematics 40 (1): 149. 
  3. ^ a b c d e f g h E.A. Jeffery (1978). "Component Minimization of the Bargman–Wigner wavefunction". Australian Journal of Physics (Melbourne: CSIRO).  NB: The convention for the four gradient in this article is μ = (∂/∂t, ∇ ), same as the Wikipedia article. Jeffery's conventions are different: μ = (−i∂/∂t, ∇ ). Also Jeffery uses collects the x and y components of the momentum operator: p± = p1 ± ip2 = px ± ipy. The components p± are not to be confused with ladder operators; the factors of ±1, ±i occur from the gamma matrices.
  4. ^ a b c d R.K Loide, I.Ots, R. Saar (2001). "Generalizations of the Dirac equation in covariant and Hamiltonian form". Journal of Physics A: Mathematical and General (Tallinn, Estonia: IoP). 
  5. ^ C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). p. 1514. ISBN 0-07-051400-3. 
  6. ^ H. Shi-Zhong, R. Tu-Nan, W. Ning, Z. Zhi-Peng (2002). "Wavefunctions for Particles with Arbitrary Spin". Beijing, China: International Academic Publishers. 
  7. ^ a b c T. Jaroszewicz, P.S Kurzepa (1992). "Geometry of spacetime propagation of spinning particles". Annals of Physics (California, USA). 
  8. ^ C.R. Hagen (1970). "The Bargmann–Wigner method in Galilean relativity". Communications in Mathematical Physics 18 (2). pp. 97–108. doi:10.1007/BF01646089. 
  9. ^ Cédric Lorcé (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 1 − Electromagnetic Current and Multipole Decomposition". arXiv:0901.4199.
  10. ^ Cédric Lorcé (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 2 − Natural Moments and Transverse Charge Densities". arXiv:0901.4200.
  11. ^ a b c Weinberg, S. (1964). "Feynman Rules for Any spin". Phys. Rev. 133 (5B): B1318–B1332. Bibcode:1964PhRv..133.1318W. doi:10.1103/PhysRev.133.B1318. ; Weinberg, S. (1964). "Feynman Rules for Any spin. II. Massless Particles". Phys. Rev. 134 (4B): B882–B896. Bibcode:1964PhRv..134..882W. doi:10.1103/PhysRev.134.B882. ; Weinberg, S. (1969). "Feynman Rules for Any spin. III". Phys. Rev. 181 (5): 1893–1899. Bibcode:1969PhRv..181.1893W. doi:10.1103/PhysRev.181.1893. 
  12. ^ Gábor Zsolt Tóth (2012). "Projection operator approach to the quantization of higher spin fields". arXiv:1209.5673.
  13. ^ V.V. Dvoeglazov (2003). "Generalizations of the Dirac Equation and the Modified Bargmann–Wigner Formalism". arXiv:hep-th/0208159.
  14. ^ D. Shay (1968). "A Lagrangian formulation of the Joos–Weinberg wave equations for spin-j particles". Il Nuovo Cimento A 57 (2): 210–218. Bibcode:1968NCimA..57..210S. doi:10.1007/BF02891000. 
  15. ^ E. Abers (2004). Quantum Mechanics. Addison Wesley. ISBN 9780131461000. 
  16. ^ a b K. Masakatsu (2012). "Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation". arXiv:1208.0644.pdf.
  17. ^ Weinberg, S (2002), "5", The Quantum Theory of Fields, vol I, ISBN 0-521-55001-7 
  18. ^ M.A. Rodriguez (1984). "Some results about the relationship between Bargmann–Wigner and Gelfand–Yaglom equations". Reports on Mathematical Physics (Madrid, Spain: Elsevier). Bibcode:1986RpMP...23....9R. doi:10.1016/0034-4877(86)90063-7. 
  19. ^ D. S. Kaparulin, S. L. Lyakhovich, A. A. Sharapov (2012). "Lagrange Anchor for Bargmann–Wigner equations". arXiv:1210.2134.pdf. Bibcode 2012arXiv1210.2134K.

Further reading[edit]

Books[edit]

  • Weinberg, S, The Quantum Theory of Fields, vol II 
  • Weinberg, S, The Quantum Theory of Fields, vol III 
  • R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1. 

Selected papers[edit]

External links[edit]

Relativistic wave equations:

Lorentz groups in relativistic quantum physics: