Galileon Equations

David Fairlie( with Thom Curtright)

Department of Mathematical Sciences, Durham University

Fort Lauderdale

December 2011

Outline

• Galilei invariant equations• Euler- Lagrange Hierarchy• Solutions• Applications and Connections

Outline

• Galilei invariant equations• Euler- Lagrange Hierarchy• Solutions• Applications and Connections

Outline

• Galilei invariant equations• Euler- Lagrange Hierarchy• Solutions• Applications and Connections

Outline

• Galilei invariant equations• Euler- Lagrange Hierarchy• Solutions• Applications and Connections

Equations which involve only second order derivatives areclearly invariant under Galileon transformations

φ(x) 7→ φ(x) + c + vµxµ

Surprisingly;it is non trivial to construct such equations whichhave a Lagrangian interpretation A few members which havethe property are•

E1 = φµµ = 0,

•E2 = φµµφνν − φ2

µν = 0,

•E3 = φλλφµµφνν − 3φλλφ2

µν + 2φλµφµνφνλ = 0.

Equations which involve only second order derivatives areclearly invariant under Galileon transformations

φ(x) 7→ φ(x) + c + vµxµ

Surprisingly;it is non trivial to construct such equations whichhave a Lagrangian interpretation A few members which havethe property are•

E1 = φµµ = 0,

•E2 = φµµφνν − φ2

µν = 0,

•E3 = φλλφµµφνν − 3φλλφ2

µν + 2φλµφµνφνλ = 0.

Equations which involve only second order derivatives areclearly invariant under Galileon transformations

φ(x) 7→ φ(x) + c + vµxµ

Surprisingly;it is non trivial to construct such equations whichhave a Lagrangian interpretation A few members which havethe property are•

E1 = φµµ = 0,

•E2 = φµµφνν − φ2

µν = 0,

•E3 = φλλφµµφνν − 3φλλφ2

µν + 2φλµφµνφνλ = 0.

Equations which involve only second order derivatives areclearly invariant under Galileon transformations

φ(x) 7→ φ(x) + c + vµxµ

Surprisingly;it is non trivial to construct such equations whichhave a Lagrangian interpretation A few members which havethe property are•

E1 = φµµ = 0,

•E2 = φµµφνν − φ2

µν = 0,

•E3 = φλλφµµφνν − 3φλλφ2

µν + 2φλµφµνφνλ = 0.

The equation En is obtained by iterating the Euler operator Eacting iteratively on the Lagrangian

L1 =

(∂φ

∂xµ

)2

multiplied by the previous equation of motion, En−1.

E = − ∂

∂φ+ ∂µ

∂

∂φµ− ∂µ∂ν

∂

∂φij. . .

(In principle the iteration continues indefinitely but it is sufficientfor our purposes to terminate at the stage of second derivativesφij , since it turns out that the iterations do not introduce anyderivatives higher than the second)

The equation En is obtained by iterating the Euler operator Eacting iteratively on the Lagrangian

L1 =

(∂φ

∂xµ

)2

multiplied by the previous equation of motion, En−1.

E = − ∂

∂φ+ ∂µ

∂

∂φµ− ∂µ∂ν

∂

∂φij. . .

(In principle the iteration continues indefinitely but it is sufficientfor our purposes to terminate at the stage of second derivativesφij , since it turns out that the iterations do not introduce anyderivatives higher than the second)

The sequence of Lagrangians is then•

L1 =

(∂φ

∂xµ

)2

•

L2 =

(∂φ

∂xµ

)2

E1 = φ2µφνν

•

L3 =

(∂φ

∂xµ

)2

E2 = φµ2(φλλφνν − φ2

λν)

etc....This sequence does not continue indefinitely, but terminateswhen there are d dimensions in Ld as Ed+1 = 0.

The sequence of Lagrangians is then•

L1 =

(∂φ

∂xµ

)2

•

L2 =

(∂φ

∂xµ

)2

E1 = φ2µφνν

•

L3 =

(∂φ

∂xµ

)2

E2 = φµ2(φλλφνν − φ2

λν)

etc....This sequence does not continue indefinitely, but terminateswhen there are d dimensions in Ld as Ed+1 = 0.

The sequence of Lagrangians is then•

L1 =

(∂φ

∂xµ

)2

•

L2 =

(∂φ

∂xµ

)2

E1 = φ2µφνν

•

L3 =

(∂φ

∂xµ

)2

E2 = φµ2(φλλφνν − φ2

λν)

etc....This sequence does not continue indefinitely, but terminateswhen there are d dimensions in Ld as Ed+1 = 0.

1. In any of these hierarchies, the Lagrangian at any givenlevel – except of course at the first level – is essentiallyproportional to the equations of motion of the Lagrangianat the previous level (hence the name Euler hierarchies).

2. The proportionality factor mentioned is essentially the veryfirst Lagrangian in the hierarchy.

3. In any of these hierarchies, Lagrangians depend on fieldsonly through their first and second derivatives, but not onderivatives of higher order nor on the fields themselves.The first Lagrangian only depends on first derivatives ofthe fields. The dependence of each of the otherLagrangians on second derivatives is multilinear.

4. All these hierarchies are finite, i.e. the iterative procedureimplied terminates after a number of steps.

5. For each hierarchy, the last non trivial equations of motionare universal, namely, up to a factor, they are independentof the initial Lagrangian.

1. In any of these hierarchies, the Lagrangian at any givenlevel – except of course at the first level – is essentiallyproportional to the equations of motion of the Lagrangianat the previous level (hence the name Euler hierarchies).

2. The proportionality factor mentioned is essentially the veryfirst Lagrangian in the hierarchy.

3. In any of these hierarchies, Lagrangians depend on fieldsonly through their first and second derivatives, but not onderivatives of higher order nor on the fields themselves.The first Lagrangian only depends on first derivatives ofthe fields. The dependence of each of the otherLagrangians on second derivatives is multilinear.

4. All these hierarchies are finite, i.e. the iterative procedureimplied terminates after a number of steps.

5. For each hierarchy, the last non trivial equations of motionare universal, namely, up to a factor, they are independentof the initial Lagrangian.

1. In any of these hierarchies, the Lagrangian at any givenlevel – except of course at the first level – is essentiallyproportional to the equations of motion of the Lagrangianat the previous level (hence the name Euler hierarchies).

2. The proportionality factor mentioned is essentially the veryfirst Lagrangian in the hierarchy.

3. In any of these hierarchies, Lagrangians depend on fieldsonly through their first and second derivatives, but not onderivatives of higher order nor on the fields themselves.The first Lagrangian only depends on first derivatives ofthe fields. The dependence of each of the otherLagrangians on second derivatives is multilinear.

4. All these hierarchies are finite, i.e. the iterative procedureimplied terminates after a number of steps.

5. For each hierarchy, the last non trivial equations of motionare universal, namely, up to a factor, they are independentof the initial Lagrangian.

1. In any of these hierarchies, the Lagrangian at any givenlevel – except of course at the first level – is essentiallyproportional to the equations of motion of the Lagrangianat the previous level (hence the name Euler hierarchies).

2. The proportionality factor mentioned is essentially the veryfirst Lagrangian in the hierarchy.

3. In any of these hierarchies, Lagrangians depend on fieldsonly through their first and second derivatives, but not onderivatives of higher order nor on the fields themselves.The first Lagrangian only depends on first derivatives ofthe fields. The dependence of each of the otherLagrangians on second derivatives is multilinear.

4. All these hierarchies are finite, i.e. the iterative procedureimplied terminates after a number of steps.

5. For each hierarchy, the last non trivial equations of motionare universal, namely, up to a factor, they are independentof the initial Lagrangian.

1. In any of these hierarchies, the Lagrangian at any givenlevel – except of course at the first level – is essentiallyproportional to the equations of motion of the Lagrangianat the previous level (hence the name Euler hierarchies).

2. The proportionality factor mentioned is essentially the veryfirst Lagrangian in the hierarchy.

3. In any of these hierarchies, Lagrangians depend on fieldsonly through their first and second derivatives, but not onderivatives of higher order nor on the fields themselves.The first Lagrangian only depends on first derivatives ofthe fields. The dependence of each of the otherLagrangians on second derivatives is multilinear.

4. All these hierarchies are finite, i.e. the iterative procedureimplied terminates after a number of steps.

5. For each hierarchy, the last non trivial equations of motionare universal, namely, up to a factor, they are independentof the initial Lagrangian.

Specifically, hierarchies with these properties were shown toexist in the following cases:

• a single field φ in d dimensions, with an arbitrary initialLagrangian (function of first derivatives only),

• A single field φ in d + 1 dimensions, the initial Lagrangiannow being an arbitrary homogeneous function of itsarguments of weight one; i.e.∑

φµ∂

∂φµL(φν) = L

Specifically, hierarchies with these properties were shown toexist in the following cases:

• a single field φ in d dimensions, with an arbitrary initialLagrangian (function of first derivatives only),

• A single field φ in d + 1 dimensions, the initial Lagrangiannow being an arbitrary homogeneous function of itsarguments of weight one; i.e.∑

φµ∂

∂φµL(φν) = L

Specifically, hierarchies with these properties were shown toexist in the following cases:

• a single field φ in d dimensions, with an arbitrary initialLagrangian (function of first derivatives only),

• A single field φ in d + 1 dimensions, the initial Lagrangiannow being an arbitrary homogeneous function of itsarguments of weight one; i.e.∑

φµ∂

∂φµL(φν) = L

Specifically, hierarchies with these properties were shown toexist in the following cases:

• a single field φ in d dimensions, with an arbitrary initialLagrangian (function of first derivatives only),

• A single field φ in d + 1 dimensions, the initial Lagrangiannow being an arbitrary homogeneous function of itsarguments of weight one; i.e.∑

φµ∂

∂φµL(φν) = L

Specifically, hierarchies with these properties were shown toexist in the following cases:

• a single field φ in d dimensions, with an arbitrary initialLagrangian (function of first derivatives only),

• A single field φ in d + 1 dimensions, the initial Lagrangiannow being an arbitrary homogeneous function of itsarguments of weight one; i.e.∑

φµ∂

∂φµL(φν) = L

The hierarchies associated with these two cases terminateafter d steps, with the following universal equations:

1.det |φµν | = 0

This is the Hessian, or Monge-Amper̀e equation, aGalileon.

2.

det∣∣∣∣ 0 φνφµ φµν

∣∣∣∣ = 0

This is the so-called Universal Field Equation, (UFE) orbordered determinant

The hierarchies associated with these two cases terminateafter d steps, with the following universal equations:

1.det |φµν | = 0

This is the Hessian, or Monge-Amper̀e equation, aGalileon.

2.

det∣∣∣∣ 0 φνφµ φµν

∣∣∣∣ = 0

This is the so-called Universal Field Equation, (UFE) orbordered determinant

The Galileons arise from the special case L1 = (φµ)2. Theymay be read off as coefficients of the characteristic polynomialof the the matrix det |φµν | = 0,; for example, consider thecharactersistic polynomial in the 4x4 case;

det

∣∣∣∣∣∣∣∣∣∣λ4 λ3 λ2 λ 1T4 T3 T2 T1 4T3 T2 T1 3 0T2 T1 2 0 0T1 1 0 0 0

∣∣∣∣∣∣∣∣∣∣where Tn = trace(T n) and Tµν = φµν Then

E1 = T1, E2 = T 21 − T2, E3 = T 3

1 − 3T1T2 + 2T3

Legendre Transform

The standard form for a Legendre transformation φ←→ Φ isgiven by

φ (x) + Φ (X ) =n∑

α=1

xαXα ,

Xα (x) =∂φ (x)

∂xα≡ ∂αφ , xα (X ) =

∂Φ (X )

∂Xα≡ ∇αΦ .

1√det (∂∂φ)

1k !Ek (∂∂φ) =

n dim

1√det (∇∇Φ)

1(n − k)!

En−k (∇∇Φ) .

That is to say, field equations for φ and Φ are related by theLegendre transform, and so are their solutions.

Solutions

In particular, the solutions of E2 = 0 in 3 dimensions areconstructed by a Legendre transform of those of E1 = ∇2φ = 0,i.e. harmonic functions. It is not easy to effect thistransformation in practise. One case where it can be doneeasily is the case of spherical symmetry, where

φ(x , y , z) =1√

(x2 + y2 + z2), Φ(X ,Y ,Z ) = (X 2 + Y 2 + Z 2)

14

In general dimension all equations Ek = 0 admit solutions whichare simultaneous solutions of the equations

∂2φ

∂x2µ

= 0,(∂φ

∂xµ

)2

This follows because of the hierarchy property of theLagrangians;

Lk =

(∂φ

∂xµ

)2

Ek−1

Both φ2µ = 0 and φµµ = 0 together give a solution to all Ek = 0

Note that this class of solutions in the case of E2 = 0 orφµµφνν = φ2

µν , makes both sides of the equation vanishindependently. A solution which does not possess this propertyis the cylindrically symmetric expression

φ =

(√x2 + y2 − iz

)2

This follows because of the hierarchy property of theLagrangians;

Lk =

(∂φ

∂xµ

)2

Ek−1

Both φ2µ = 0 and φµµ = 0 together give a solution to all Ek = 0

Note that this class of solutions in the case of E2 = 0 orφµµφνν = φ2

µν , makes both sides of the equation vanishindependently. A solution which does not possess this propertyis the cylindrically symmetric expression

φ =

(√x2 + y2 − iz

)2

Self Dual solutionsSuppose we impose the ansatz;

xµV (φ)µ = 1.

(Here the subscript µ is a vector index, not a derivative.) Then

∂φ

∂xµ= − Vµ∑

xµV ′µ

∂2φ

∂xµν= −

(VµV ′ν + VνV ′µ)

(∑

xαV ′α)2)+

VµVν(∑

xβV ′′β )

(∑

xαV ′α)3)

Inserting this expression into E2 = 0 we obtain

1(∑

xαV ′α)4

((∑

VµVµ)(∑

V ′νV ′ν)− (∑

VµV ′µ)2)

By construction solutions of this type will inevitably be complexin Euclidean space, but may be real in Minkowsky space. Thisis clearly zero if the constraint

∑VµVµ = 0 is imposed. This

solution also works in any dimension.

Moreover, it is easy to see that this solution also solves∇2φ = 0, the Legendre transform of E2 = 0 in 3 dimensions.The constraint in terms of the field is simply

∑(∂φ

∂xµ

)2

= 0.

This class of solutions is invariant under conformaltransformations of the independent variables. This ansatz alsosolves En = 0, for n > 2. Another remarkable property of thissolution is that if φ is replaced by any function of φ it remains asolution of these equations. Not all solutions satisfy thiscondition, however. A counterexample is φ =

√x1x2 · · · xd .

ApplicationsIt is tempting to view the sequence of equations En = 0 aspotential generalisations in the spirit of the MOND suggestedexplanations of Dark Matter, providing corrections to theNewtonian potential. Take

E1 = λE2 .

The static, spherically symmetric solutions in this case satisfy

− 1r2∂r

(r2∂rφ

)=

2λr2 ∂r

(r (∂rφ)2

),

which has an immediate first integral, and solution,

∂rφ =r

4λ

(−1±

√1 +

8λC1

r3

).

Integrating this gives a hypergeometric function.

ApplicationsIt is tempting to view the sequence of equations En = 0 aspotential generalisations in the spirit of the MOND suggestedexplanations of Dark Matter, providing corrections to theNewtonian potential. Take

E1 = λE2 .

The static, spherically symmetric solutions in this case satisfy

− 1r2∂r

(r2∂rφ

)=

2λr2 ∂r

(r (∂rφ)2

),

which has an immediate first integral, and solution,

∂rφ =r

4λ

(−1±

√1 +

8λC1

r3

).

Integrating this gives a hypergeometric function.

For small r the solution behaves like C0 + C1√

r and therefore itis not singular at the origin. On the other hand, for large r ,upon taking the upper + sign in the previous equation thesolution behaves like C0 + C1

1r while the lower − sign choice

gives a solution that grows like r2 for large r .

In the case of E2, the expression for the equation can bedecomposed into a sum of 2x2 Monge-Amper̀e (Hessian)equations;

E2 = φxµxµφxνxν − (φxµxν )2 =∑

over all pairs µ,ν

Hessian (φ(xµ, xν))

Similarly for 3x3 Hessians,

E3 =∑

all perms

Hessian (φ(xµ, xν , xλ))

Similar remarks hold for the Universal field equationgeneralisation to more dimensions.

Outlook

• Homogeneous equations of second order which admit aLagrangian derivation are special.

E2 = φµµφνν − φ2µν = 0,

The relative coefficient is fixed.• Large classes of solutions have been found, but a

comprehensive catalogue is known only for E1 = φµµ = 0,,(the D’alembertian), and En in n dimensions (the Hessian,or Monge Amper̀e equation)

• Linear combinations of the Galileon equations may give amore theoretical understanding of modifications of gravity,which come into the category of dark matter.

Outlook

• Homogeneous equations of second order which admit aLagrangian derivation are special.

E2 = φµµφνν − φ2µν = 0,

The relative coefficient is fixed.• Large classes of solutions have been found, but a

comprehensive catalogue is known only for E1 = φµµ = 0,,(the D’alembertian), and En in n dimensions (the Hessian,or Monge Amper̀e equation)

• Linear combinations of the Galileon equations may give amore theoretical understanding of modifications of gravity,which come into the category of dark matter.

Outlook

• Homogeneous equations of second order which admit aLagrangian derivation are special.

E2 = φµµφνν − φ2µν = 0,

The relative coefficient is fixed.• Large classes of solutions have been found, but a

comprehensive catalogue is known only for E1 = φµµ = 0,,(the D’alembertian), and En in n dimensions (the Hessian,or Monge Amper̀e equation)

• Linear combinations of the Galileon equations may give amore theoretical understanding of modifications of gravity,which come into the category of dark matter.

References

D.B. Fairlie, J. Govaerts and A. Morozov, Nuclear Physics B373(1992) 214-232.

D.B. Fairlie, J. Govaerts, Physics Letters 281B (1992) 49-53.

D.B. Fairlie, J. Govaerts, J. Math Phys33 (1992) 3543-3566.

D.B.Fairlie, Comments on Galileons, J.PhysA44:305201 (2011),arXiv:1102.1594 [hep-th].

References

D.B. Fairlie, J. Govaerts and A. Morozov, Nuclear Physics B373(1992) 214-232.

D.B. Fairlie, J. Govaerts, Physics Letters 281B (1992) 49-53.

D.B. Fairlie, J. Govaerts, J. Math Phys33 (1992) 3543-3566.

D.B.Fairlie, Comments on Galileons, J.PhysA44:305201 (2011),arXiv:1102.1594 [hep-th].

References

D.B. Fairlie, J. Govaerts and A. Morozov, Nuclear Physics B373(1992) 214-232.

D.B. Fairlie, J. Govaerts, Physics Letters 281B (1992) 49-53.

D.B. Fairlie, J. Govaerts, J. Math Phys33 (1992) 3543-3566.

D.B.Fairlie, Comments on Galileons, J.PhysA44:305201 (2011),arXiv:1102.1594 [hep-th].

References

D.B. Fairlie, J. Govaerts and A. Morozov, Nuclear Physics B373(1992) 214-232.

D.B. Fairlie, J. Govaerts, Physics Letters 281B (1992) 49-53.

D.B. Fairlie, J. Govaerts, J. Math Phys33 (1992) 3543-3566.

D.B.Fairlie, Comments on Galileons, J.PhysA44:305201 (2011),arXiv:1102.1594 [hep-th].