aa r X i v : . [ m a t h . A P ] M a y Matrix Gauge FieldsandNoether’s Theorem
J. de GRAAF
Eindhoven University of Technology, Mathematics,Casa Reports 14-14, May 2014Preface and Summary
These notes are about systems of 1st and 2nd order (non-)linear partial differential equa-tions which are formed from a
Lagrangian density L ψ : R N → C ,Symbolically : x L ψ ( x ) = L ( ψ ( x ) ; ∇ ψ ( x ) ; x ) , by means of the usual Euler-Lagrange variational rituals. The non subscripted L will denotethe ’ proto-Lagrangian ’, which is a function of a finite number of variables: L : C r × c × C Nr × c × R N → C . In this L one has to substitute matrix-valued functions ψ : R N → C r × c and ∇ ψ : R N → C Nr × c for obtaining the Lagrangian density L ψ . In our considerations the role and thespecial properties of the proto-Lagrangian L are crucial.These notes have been triggered by physicist’s considerations: (1) on obtaining the ’classi-cal’, that is the ’pre-quantized’, wave equations for matter fields from variational principles, (2) on conservation laws and (3) on ’gauge field extensions’. For the humble mathematicalanthropologist the rituals in physics textbooks have not much changed during the last fourdecades. Neither have they become much clearer. Compare e.g. [DM] and [W].The underlying notes give special attention to the following • In expressions (=’equations’) for Lagrange densities often both ψ and its hermiteantransposed ψ † appear. Are they meant as independent variables or not? Mostly,from the context the suggestion arises that ’variation’ of ψ and ’variation’ of ψ † lead1o the same Euler-Lagrange equations. Why? Our remedy is doubling the matrixentries in the proto-Lagrangian and thereby making the Lagrangian density explicitlydependent on both ψ , ψ † and their derivatives: So for L ψ ( x ) we take expressions like L ψ ( x ) = L ( ψ ( x ) ; ψ ( x ) † ; ∇ ψ ( x ) ; ∇ ψ ( x ) † ; x ) . A suitable condition is then that the Lagrangian functional L [ ψ ] = Z R N L ψ ( x ) d x only takes real values (Thm 2.4). • For ’free gauge fields’ the situation is somewhat different. Now the dependent varia-bles, named A µ , ≤ µ ≤ N , take their values in some fixed Lie-algebra g ⊂ C c × c .Although g mostly contains complex matrices it is a real vector space in interestingcases. (Note that u (1) = i R is a real vector space!). Therefore it needs a separatetreatment. • The traditional conservation laws for quantities like energy, momentum, moment ofmomentum, . . . , turn out to be based on
External Infinitesimal Symmetries of theproto-Lagrangian. This means the existence of a couple of linear mappings
K : C r × c → C r × c , L : C Nr × c → C Nr × c , together with an affine mapping x
7→ − sa + e sA x , such that for all matrices P ∈ C r × c , Q ∈ C Nr × c and x ∈ R N , L ( e s K P ; e s L Q ; − sa + e sA x ) = L (P , Q ; x ) + O ( s ) . Of course the presented conservation laws are just special cases of Noether’s Theorem. • For the construction of gauge theories one needs, in physicist’s terminology, a ’glo-bal symmetry of the Lagrangian’. To achieve this, an Internal Symmetry of theproto-Lagrangian L is required here: For some fixed Lie-group G ⊂ C c × c , the proto-Lagrangian satisfies L (P U ; Q U ; x ) = L (P ; Q ; x ) , for all P ∈ C r × c , Q ∈ C Nr × c , U ∈ G , x ∈ R N . Roughly speaking, a gauge theory for a Lagrangian based system of PDE’s is somekind of symmetry preserving extension of the original Lagrangian density with new(dependent) ’field’-variables x A ( x ) = [ A ( x ) , . . . , A N ( x )] on R N added, such thatthe original ’quantities’ ψ become subjected to the ’gauge fields’ A and viceversa.Since about a century, Weyl 1918, it is well known that, given the existence of some’global symmetry group’ G of L , an extension of type L ψ,A ( x ) = L ( ψ ; ∇ ψ + ψ · A ; x ) + G ( A ; ∇ A ; x ) , is often possible. This extension has to exhibit what physicists call, a ’Local Symme-try’ : The Lagrangian density remains unaltered if in L ψ,A the quantities ψ and A are,each in their own way, subjected to group actions taken from G loc = C ∞ ( R N ; G ) ,2hich is the group of smooth maps R N → G . The added ’gauge fields’ A have totake their values in the Lie Algebra g of the symmetry group G .Summarizing, ’locally symmetric’ means, symbolically, L (cid:0) ψU ; ∇ ( ψU )+( ψU ) · ( A ⊳ U ) ; x )+ G ( A ⊳ U ; ∇ ( A ⊳ U ) ; x (cid:1) == L (cid:0) ψ ; ∇ ψ + ψ · A ; x (cid:1) + G (cid:0) A ; ∇ A ; x (cid:1) , for all U ∈ G loc . • The considerations in the underlying notes not only include the standard hyperbolicevolution equations of pre-quantized fields. Wide classes of parabolic/elliptic systemsturn out to have gauge extensions as well. Note the subtle extra condition (5.14) inThm 5.5 which is, besides internal symmetry of the proto-Lagrangian, necessary forgauge extensions. Its necessity lies in the fact that one has to reconcile the complex vector space, in which the ψ take their values, with the real vector space g , the Lie-Algebra. In the standard preludes to quantum field the requirement (5.14) is neverdiscussed, but manifestly met with. • These notes do not contain functional analysis or differential geometry. The readerwill find only bare elementary considerations on matrix-valued functions: The co-lumns of the x ψ ( x ) ∈ C r × c might describe the ’pre-quantized wave functions’ ofindividual elementary particles, whereas the ’components’ of x A ( x ) ∈ g N , with g ⊂ C c × c , might represent the pre-quantized gauge fields. For an elementary andvery readable account on the differential geometrical aspects, see the contributions3-4 in [JP]. CONTENTS1.
Foretaste: Some gauge-type calculations p.3 Stationary points of complex-valued functionals p.6 Free Gauge Fields p.13 Noether Fluxes p.19 Static/Dynamic Gauge Extensions of Lagrangians p.26 A. Addendum on Free Gauge Fields p.34 B. Electromagnetism p.35References p.363
Foretaste: Some gauge-type calculations
For functions Ψ : R N → C r × c we consider, by way of example, the PDE Γ µ (cid:0) ∂ µ Ψ + Ψ A µ (cid:1) + M Ψ = f, (1.1)with prescribed matrix valued coefficients Γ µ : R N → C r × r , A µ : R N → C c × c , ≤ µ ≤ N, M : R N → C r × r , and prescribed right hand side f : R N → C r × c . All considered functions are supposed tobe sufficiently smooth. The summation convention for upper and lower indices applies.In physics each column of Ψ may represent a ’classical-particle wave’. The A µ may thenrepresent ’gauge fields’. Theorem 1.1
Let U , V : R N → C c × c and suppose them invertible with U − , V − : R N → C c × c .The function ˆ Ψ = Ψ U : R N → C r × k , with Ψ any solution of (1.1) is a solution of Γ µ (cid:0) ∂ µ ˆ Ψ + ˆ Ψ ˆ A µ (cid:1) + M ˆ Ψ = ˆ f , (1.2) if and only if we take the new coefficients ˆ A µ = U − A µ U − U − ( ∂ µ U ) and ˆ f = f U .In addition we have ˆˆ A µ = ( U V ) − A µ ( U V ) − ( U V ) − ( ∂ µ ( U V )) = V − ˆ A µ V − V − ( ∂ µ V ) . Proof:
Multiply (1.1) from the right by U and rearrange. (cid:4) In the next Theorem a ’transformation property’ for matrix valued functions is derived.
Theorem 1.2
Let A µ : R N → C c × c and ˆ A µ = U − A µ U − U − ( ∂ µ U ) . Define F µν = ∂ µ A ν − ∂ ν A µ − (cid:0) A µ A ν − A ν A µ (cid:1) . (1.3) Then ˆ F µν = ∂ µ ˆ A ν − ∂ ν ˆ A µ − (cid:0) ˆ A µ ˆ A ν − ˆ A ν ˆ A µ (cid:1) = U − F µν U . (1.4) Proof:
First note that from ∂ µ ( U − U ) = ∂ µ I = 0 it follows that ∂ µ ( U − ) = −U − ( ∂ µ U ) U − .Calculate ∂ µ ˆ A ν = ∂ µ (cid:0) U − A ν U − U − ( ∂ ν U ) (cid:1) == U − ( ∂ µ A ν ) U − U − ( ∂ µ U ) U − A ν U + U − A ν ( ∂ µ U ) + U − ( ∂ µ U ) U − ( ∂ ν U ) − U − ( ∂ µ ∂ ν U ) . and ˆ A µ ˆ A ν = (cid:8) U − A µ U − U − ( ∂ µ U ) (cid:9)(cid:8) U − A ν U − U − ( ∂ ν U ) (cid:9) == U − (cid:0) A µ A ν (cid:1) U − (cid:0) U − A µ U (cid:1)(cid:0) U − ( ∂ ν U ) (cid:1) − (cid:0) U − ( ∂ µ U ) (cid:1)(cid:0) U − A ν U (cid:1) + (cid:0) U − ( ∂ µ U ) (cid:1)(cid:0) U − ( ∂ ν U ) (cid:1) . Interchange the indices for two more terms and add according to (1.4). All rubbish termscancel out. (cid:4)
We now look for sesqui-linear conservation laws which hold for suitable classes of A µ ondition 1.3 K : R N → C r × r , is such that i : K Γ µ = ( K Γ µ ) † , ii : ∂ µ ( K Γ µ ) = 0 , iii : KM + M † K † = 0 .Here, the dagger † denotes ’Hermitean transposition’.Note that in the important special case that Γ µ = (Γ µ ) † , Γ µ is constant and M = − M † ,the condition is satisfied by K = I , the identity matrix. In the case of the Dirac equationone could take K = Γ . Cf. [M], Messiah II pp. 890-899. Theorem 1.4
Let K : R N → C r × r satisfy Condition 1.3.Fix some J ∈ C c × c .Let A µ : R N → C c × c satisfy A † µ J + J A µ = 0 , ≤ µ ≤ N .Let U : R N → C c × c satisfy U † ( x ) J U ( x ) = J , x ∈ R N . a. For any solution Ψ of (1.1) with f = 0 , there is the conservation law N X µ =1 ∂ µ Tr (cid:0) J − [ Ψ † K Γ µ Ψ ] (cid:1) = 0 . (1.5) b. This conservation law is a gauge invariant local conservation law .That means Tr (cid:0) J − [ ˆ Ψ † K Γ µ ˆ Ψ ] (cid:1) = Tr (cid:0) J − [ Ψ † K Γ µ Ψ ] (cid:1) , ≤ µ ≤ N . Proofa.
Take f = 0 in (1.1)and multiply from the left with Ψ † K : Ψ † K Γ µ (cid:0) ∂ µ Ψ (cid:1) + Ψ † K Γ µ Ψ A µ + Ψ † KM Ψ = 0 . (1.6)The Hermitean transpose reads (cid:0) ∂ µ Ψ (cid:1) † ( K Γ µ ) † Ψ + A † µ Ψ † ( K Γ µ ) † Ψ + Ψ † M † K † Ψ = 0 . (1.7)Multiply (1.6) from the right with J − and (1.7) from the left with J − . Add those twoidentities and take the trace. Use Condition 1.3 and the properties Tr( AB ) = Tr( BA ) , Tr( A + B ) = Tr( A ) + Tr( B ) and ∂ µ Tr( A ) = Tr( ∂ µ A ) . The sum of the 1st terms of (1.6),(1.7) result in Tr (cid:8) J − (cid:2) Ψ † ( K Γ µ ) ∂ µ Ψ + ( ∂ µ Ψ ) † ( K Γ µ ) † Ψ (cid:3)(cid:9) == ∂ µ Tr (cid:8) J − Ψ † ( K Γ µ ) Ψ (cid:9) − Tr (cid:8) J − Ψ † ∂ µ ( K Γ µ ) Ψ (cid:9) = ∂ µ Tr (cid:8) J − Ψ † ( K Γ µ ) Ψ (cid:9) . The sum of the 2nd terms of (1.6), (1.7) is Tr (cid:8) Ψ † K Γ µ Ψ (cid:0) A µ J − + J − A † µ (cid:1)(cid:9) = 0 . In the non-covariant form, i.e. the original form, of Dirac’s equation one has Γ = I, Γ κ = γ γ κ , ≤ κ ≤ , where the γ µ , ≤ µ ≤ are Dirac-Clifford matrices, which make the Dirac equation covariantproof. Tr (cid:8) J − Ψ † ( KM + M † K † ) Ψ (cid:9) = 0 . Thus, we find (1.5) b. By putting hats on Ψ and A µ our considerations can be rephrased for PDE (1.2).Remind that from U † J U = J it follows that J − U † = U − J − . Finally Tr (cid:0) J − U † [ Ψ † K Γ µ Ψ ] U (cid:1) = Tr (cid:0) U − J − [ Ψ † K Γ µ Ψ ] U (cid:1) = Tr (cid:0) J − [ Ψ † K Γ µ Ψ ] (cid:1) . (cid:4) In this section we pay some attention to the Euler Lagrange field equations in the com-plex field case. Most physics textbooks start, in a rather verbose way, with 18th centuryvariational rituals. However most of them become suddenly very vague, or fall completelysilent, when state functions involving complex variables come into play! In order to getsome feeling for such Lagrangians, we first mention a finite dimensional toy result.
Theorem 2.1
Let f : C n × C n ∋ ( z ; w ) f ( z, w ) ∈ C be an analytic function of n complex variables with the special property f ( z, z ⋆ ) ∈ R , forall z ∈ C n . Here z = x + i y , z ⋆ = x − i y . a. Consider the function R n × R n ∋ ( x ; y ) g ( x, y ) = f ( z, z ⋆ ) = f ( x + i y, x − i y ) ∈ R . The relations between the (real) partial derivatives of g at ( x, y ) and the (complex) partialderivatives of f at ( z, z ⋆ ) are ∂g∂x ( x, y ) = ∂f∂z ( z, z ⋆ ) + ∂f∂w ( z, z ⋆ ) ∂f∂z ( z, z ⋆ ) = 12 (cid:0) ∂g∂x ( x, y ) − i ∂g∂y ( x, y ) (cid:1) ∂g∂y ( x, y ) = i ∂f∂z ( z, z ⋆ ) − i ∂f∂w ( z, z ⋆ ) ∂f∂w ( z, z ⋆ ) = 12 (cid:0) ∂g∂x ( x, y ) + i ∂g∂y ( x, y ) (cid:1) (2.1) ∂f∂w ( z, z ⋆ ) = ∂f∂z ( z, z ⋆ ) . For g to have a stationary point at ( a ; b ) ∈ R n × R n each one of the following threeconditions is necessary and sufficient • ∂g∂x ( a, b ) = ∂g∂y ( a, b ) = 0 , • ∂f∂z ( a + i b, a − i b ) = 0 , • ∂f∂w ( a + i b, a − i b ) = ” ∂f∂z ⋆ ( a + i b, a − i b ) ” = 0 . (2.2) c. If the special property f ( x + i y, x − i y ) ∈ R is relaxed to φ ( f ( x + i y, x − i y )) ∈ R forsome non-constant analytic φ : C → C , then the ’stationary point result’ b. still holds. Proof:
Straightforward calculation (cid:4)
In Theorem 2.4 an ∞ -dimensional generalisation of this result is presented. A special bookkeeping
In the sequel, for the above variable z , usually a matrix Z ∈ C r × c will be taken. In order toexplain our bookkeeping and also for some special properties, we now consider an analyticfunction of 2 matrix variables F : C r × c × C c × r → C : (Z ; W) F (Z , W) . (2.3)Because of Hartog’s Theorem, see [H] Thm 2.2.8, it is enough to assume analyticity withrespect to each entry of each matrix separately.The (complex!) partial derivatives of F are gathered in matrices, (Z; W) F ( ) (Z , W) ∈ C c × r , (Z; W) F ( ) (Z , W) ∈ C r × c , with (cid:2) F ( ) (cid:3) ij = (cid:2) ∂ F ∂ Z (cid:3) ij = ∂ F ∂ Z ji , (cid:2) F ( ) (cid:3) kℓ = (cid:2) ∂ F ∂ W (cid:3) kℓ = ∂ F ∂ W ℓk . (2.4)In our notation the C -linearization of F at (Z , W) , for ε ∈ C , | ε | small, reads F (Z + ε H , W + ε K) = F (Z , W) + ε Tr (cid:8) [ F ( ) ]H } + ε Tr (cid:8) [ F ( ) ]K (cid:9) + O ( | ε | ) . (2.5) Notation:
Sometimes, in order to avoid excessive use of brackets, it is convenient to write Tr (cid:8) F ( ) : H } instead of Tr (cid:8) [ F ( ) ]H } .Also, without warning, in proofs sometimes Einstein’s summation convention for repeatedupper and lower indices will be used. 7ext split Z in real and imaginary parts Z = X + iY and introduce the function f F : R r × c × R r × c → C : (X; Y) f F (X , Y) = F (Z , Z † ) = F (X + iY , X ⊤ − iY ⊤ ) . (2.6)The R -linearization of f F at (X , Y) for ε ∈ R , | ε | small, can now be written f F (X + ε A , Y + ε B) = f F (X , Y) + ε Tr (cid:8) ∂ f F ∂ X A + ε Tr (cid:8) ∂ f F ∂ Y B (cid:9)(cid:9) + O ( ε ) , (2.7)with Tr (cid:8) ∂ f F ∂ X A (cid:9) = Tr (cid:8) [ F ( ) ]A (cid:9) + Tr (cid:8) [ F ( ) ]A ⊤ (cid:9) = Tr (cid:8)(cid:0) [ F ( ) ] + [ F ( ) ] ⊤ (cid:1) A (cid:9) , Tr (cid:8) ∂ f F ∂ Y B (cid:9) = Tr (cid:8) i[ F ( ) ]B (cid:9) + Tr (cid:8) − i[ F ( ) ]B ⊤ (cid:9) = Tr (cid:8) i (cid:0) [ F ( ) ] − [ F ( ) ] ⊤ (cid:1) B (cid:9) , (2.8)where the matrices X , Y , A , B are all real. The (complex) derivatives F ( ) , F ( ) are takenat (Z , Z † ) . In the usual (somewhat confusing) notation, this corresponds to ∂ f F ∂ X = ∂ F ∂ X = ∂ F ∂ Z + (cid:2) ∂ F ∂ Z † (cid:3) ⊤ , ∂ f F ∂ Y = ∂ F ∂ Y = i ∂ F ∂ Z − i (cid:2) ∂ F ∂ Z † (cid:3) ⊤ , (2.9)and, similarly sloppy, ∂ F ∂ Z = 12 (cid:0) ∂ F ∂ X − i ∂ F ∂ Y (cid:1) , (cid:2) ∂ F ∂ Z † (cid:3) ⊤ = 12 (cid:0) ∂ F ∂ X + i ∂ F ∂ Y (cid:1) . (2.10)If it happens that Z F (Z , Z † ) is R -valued, the results of Theorem (2.1) can be rephrased. Theorem 2.2
Let, as in (2.3), F : C r × c × C c × r ∋ (Z; W) F (Z , W) ∈ C . be analytic . Suppose F (Z , Z † ) ∈ R , for all Z ∈ C r × c . Write Z = X + iY . Denote f F : R r × c × R r × c → R : (X; Y) f F (X , Y) = F (Z , Z † ) = F (X + iY , X ⊤ − iY ⊤ ) , • We have F ( ) (Z , Z † ) = [ F ( ) (Z , Z † )] † . (2.11) Further, for the function f F to have a stationary point at (A ; B) ∈ R r × c × R r × c each one of the following three conditions is necessary and sufficient • ∂ f F ∂ X (A , B) = ∂ f F ∂ Y (A , B) = 0 • F ( ) (A + iB , A ⊤ − iB ⊤ ) = ” ∂ F ∂ Z (A + iB , A ⊤ − iB ⊤ ) ” = 0 • F ( ) (A + iB , A ⊤ − iB ⊤ ) = ” ∂ F ∂ Z † (A + iB , A ⊤ − iB ⊤ ) ” = 0 . (2.12)8 roof: Is mostly a reformulation of the preceding theorem. It follows directly from (2.9)-(2.10). (cid:4)
In order to build the concept of
Lagrangian density we need an analytic function, named proto-Lagrangian , L : C r × c × C c × r × C Nr × c × C c × Nr × R N → C , (P; Q ⊤ ; R ; S ⊤ ; x ) L (P; Q ⊤ ; R ; S ⊤ ; x ) , (2.13)where P ∈ C r × c , R = col (cid:2) R , . . . , R N (cid:3) , R µ ∈ C r × c , ≤ µ ≤ N , Q ⊤ ∈ C c × r , S ⊤ = row (cid:2) S ⊤ , . . . , S ⊤ N (cid:3) , S ⊤ µ ∈ C c × r , ≤ µ ≤ N .
Instead of (2.13) it will be convenient sometimes to denote the proto Lagrangian by L (P; Q ⊤ ; . . . , R µ , . . . ; . . . , S ⊤ µ , . . . ; x ) . It will be required that L (O; O ⊤ ; O ; O ⊤ ; x ) = 0 .The (complex) partial derivatives of L , cf. (2.4)-(2.5), with respect to its N + 2 matrixarguments are denoted, respectively, L ( o ) , L ( o⋆ ) , L (1) , . . . , L ( N ) , L (1 ⋆ ) , . . . , L ( N⋆ ) . The (real) partial derivatives of L , with respect to the vector variable x is denoted L ( ∇ ) .For any given matrix-valued function Ψ : R N → C r × c , we define a Lagrangian density L ψ : R N → C , by substitution of Ψ , its 1st derivatives ∂ µ Ψ = Ψ , µ , ≤ µ ≤ N , and thehermitean transposed of all those, in L : x L ψ ( x ) = L ( Ψ ( x ); Ψ † ( x ); ∇ Ψ ( x ) ; ∇ Ψ † ( x ) ; x ) , (2.14)where ∇ Ψ ( x ) = col (cid:2) ∂ Ψ ( x ) , . . . , ∂ N Ψ ( x ) (cid:3) ∈ C Nr × c , ∇ Ψ † ( x ) = row (cid:2) ∂ Ψ † ( x ) , . . . , ∂ N Ψ † ( x ) (cid:3) ∈ C c × Nr . Also the matrix-valued functions x [ L ( µ ) ψ ]( x ) = [ L ( µ ) ]( Ψ ( x ); Ψ † ( x ); ∇ Ψ ( x ) ; ∇ Ψ † ( x ) ; x ) ∈ C c × r , similarly x [ L ( µ⋆ ) ψ ] ∈ C r × c , and x L ( ∇ ) ψ ∈ R N , will be used.On a suitable space of functions Ψ : R N → C r × c , it often makes sense to define the Lagrangian functional Ψ
7→ L ( Ψ , Ψ † ) = Z R N L ( Ψ ( x ); Ψ † ( x ); ∇ Ψ ( x ) ; ∇ Ψ † ( x ) ; x ) d x ∈ C . (2.15)9 emark 2.3 The Lagrangian functional L remains the same if we replace L by L ( Ψ ; Ψ † ; ∇ Ψ ; ∇ Ψ † ; x ) + ∂ µ w µ ( Ψ , Ψ † , x ) , with w µ a vectorfield which vanishes sufficiently rapidly at infinity.Therefore the functional Ψ
7→ L ( Ψ , Ψ † ) is R -valued if L ( Ψ ; Ψ † ; ∇ Ψ ; ∇ Ψ † ; x ) − L ( Ψ ; Ψ † ; ∇ Ψ ; ∇ Ψ † ; x ) = ∂ µ W µ ( Ψ , Ψ † , x ) , i.e. the divergence of a vector field. Note that L may be R -valued while L ψ is not !!If we split Ψ into real and imaginary parts: Ψ = Ψ Re + i Ψ Im and Ψ ,µ = Ψ Re ,µ + i Ψ Im ,µ ,the R -directional derivatives with respect to Ψ Re and Ψ Im of the Lagrangian functional L are explained by (cid:10) D Ψ Re L , A (cid:11) = dd ε L ( Ψ + ε A , Ψ † + ε A ⊤ ) (cid:12)(cid:12)(cid:12) ε =0 == dd ε Z R N L ( Ψ ( x ) + ε A ( x ); Ψ † ( x ) + ε A ⊤ ( x ); ∇ (cid:0) Ψ ( x ) + ε A ( x ) (cid:1) ; ∇ (cid:0) Ψ † ( x ) + ε A ⊤ ( x ) (cid:1) ; x ) d x (cid:12)(cid:12)(cid:12) ε =0 , with A : R N → R r × c , and ε ∈ R , | ε | small . (cid:10) D Ψ Im L , B (cid:11) = dd ε L ( Ψ + ε i B , Ψ † − ε i B ⊤ ) (cid:12)(cid:12)(cid:12) ε =0 == dd ε Z R N L ( Ψ ( x ) + ε i B ( x ); Ψ † ( x ) − ε i B ⊤ ( x ); ∇ (cid:0) Ψ ( x ) + ε i B ( x ) (cid:1) ; ∇ (cid:0) Ψ † ( x ) − ε i B ⊤ ( x ) (cid:1) ; x ) d x (cid:12)(cid:12)(cid:12) ε =0 , with B : R N → R r × c , and ε ∈ R , | ε | small . When calculating the C -directional derivatives D Ψ L , D Ψ † L , the variables Ψ , Ψ † are con-sidered to be independent. These derivatives are supposed to be elements in the (complex)linear dual of L ( R N ; C r × c ) . They are explained by (cid:10) D Ψ L , H (cid:11) = dd ε L ( Ψ + ε H , Ψ † ) (cid:12)(cid:12)(cid:12) ε =0 == dd ε Z R N L ( Ψ ( x ) + ε H ( x ); Ψ † ( x ); ∇ (cid:0) Ψ ( x ) + ε H ( x ) (cid:1) ; ∇ Ψ † ; x ) d x (cid:12)(cid:12)(cid:12) ε =0 , with H : R N → C r × c , and ε ∈ C , | ε | small . (cid:10) D Ψ † L , K (cid:11) = dd ε L ( Ψ , Ψ † + ε K ) (cid:12)(cid:12)(cid:12) ε =0 == dd ε Z R N L ( Ψ ( x ); Ψ † ( x ) + ε K ( x ); ∇ Ψ ( x ) ; ∇ ( Ψ † ( x ) + ε K ( x )) ; x ) d x (cid:12)(cid:12)(cid:12) ε =0 , with K : R N → C c × r , and ε ∈ C , | ε | small . For H , K , A , B vanishing sufficiently rapidly at ∞ a partial integration leads to thestandard Euler-Lagrange expressions for the functional derivatives of L .10 heorem 2.4 Assume that L is R -valued. (Cf. Remark 2.3). If Ψ satisfies any one of the followingthree Lagrangian systems D Ψ L = [ L ( o ) ψ ] − N X µ =1 ∂∂x µ [ L ( µ ) ψ ] = 0 , D Ψ † L = [ L ( o⋆ ) ψ ] − N X µ =1 ∂∂x µ [ L ( µ⋆ ) ψ ] = 0 , D Ψ Re L = ∂ L ∂ Ψ Re − N X µ =1 ∂∂x µ ∂ L ∂ Ψ Re ,µ = 0 , D Ψ Im L = ∂ L ∂ Ψ Im − N X µ =1 ∂∂x µ ∂ L ∂ Ψ Im ,µ = 0 . , (2.16) with L = L ( Ψ ( x ); Ψ † ( x ); ∇ Ψ ( x ) ; ∇ Ψ † ( x ) ; x ) , then it also satisfies the other two. Proof:
With the notation (2.8)-(2.10) we obtain ∂ L ∂ Ψ Re = L ( o ) + [ L ( o⋆ ) ] ⊤ , ∂ L ∂ Ψ Im = i L ( o ) − i[ L ( o⋆ ) ] ⊤ , (2.17)and, the other way round, (cid:2) L ( o⋆ ) (cid:3) ⊤ = 12 (cid:0) ∂ L ∂ Ψ Re + i ∂ L ∂ Ψ Im (cid:1) , L ( o ) = 12 (cid:0) ∂ L ∂ Ψ Re − i ∂ L ∂ Ψ Im (cid:1) , (2.18)and similar expressions with ( o ) , ( o⋆ ) replaced by ( µ ) , ( µ⋆ ) and Ψ , Ψ Re , Ψ Im replacedby Ψ ,µ , Ψ Re ,µ , Ψ Im ,µ . Then D Ψ L = (cid:0) D Ψ Re L − i D Ψ Im L (cid:1)(cid:2) D Ψ † L (cid:3) ⊤ = (cid:0) D Ψ Re L + i D Ψ Im L (cid:1) D Ψ Re L = D Ψ L + (cid:2) D Ψ † L (cid:3) ⊤ (cid:2) D Ψ Im L (cid:3) ⊤ = i D Ψ L − i (cid:2) D Ψ † L (cid:3) ⊤ . If we take into account that the entries of the matrix valued functions D Ψ Re L and D Ψ Im L are R -valued, we find (cid:2) D Ψ † L (cid:3) † = (cid:2) D Ψ L (cid:3) , (2.19)from which the theorem easily follows. (cid:4) Examples 2.5 (Matter Fields)a)
Let Γ µ and M be constant complex matrices with Γ µ † = Γ µ and M = − M † . Then theLagrangian density L ψ = i Tr (cid:8) Ψ † Γ µ ∂ µ Ψ + Ψ † M Ψ (cid:9) , (2.20)for Ψ : R N → C r × c , satisfies the condition of Theorem (2.4) and leads to (1.1) with A = 0 . b) Let Γ µ , ≤ µ ≤ N : R N → C r × r . Let A µ , ≤ µ ≤ N : R N → C c × c .Let M : R N → C r × r .Suppose both the existence of K : R N → C r × r , having inverse K − ( x ) , for all x ∈ R N ,and an invertible J ∈ C c × c with J † = J , such that:11 K Γ µ ) † = K Γ µ , , ≤ µ ≤ N, A † µ ( x ) J + J A µ ( x ) = 0 , ≤ µ ≤ N, x ∈ R N , and KM + M † K † − ∂ µ (cid:0) K Γ µ (cid:1) = 0 .Then the Lagrangian density L ψ = i Tr (cid:8) Ψ † K (Γ µ ∂ µ Ψ ) J − + Ψ † K (Γ µ Ψ A µ ) J − + Ψ † KM Ψ J − (cid:9) , (2.21)for Ψ : R N → C r × c satisfies L − L = ∂ µ w and hence the condition of Theorem (2.4).It leads to the ’matter-field equation’ Γ µ ∂ µ Ψ + Γ µ Ψ A µ + M Ψ = 0 (2.22) Indeed.
Taking suitable combinations we find respectively Tr (cid:8) Ψ † K Γ µ ( ∂ µ Ψ ) J − + J − ( ∂ µ Ψ ) † ( K Γ µ ) † Ψ ) (cid:9) = Tr (cid:8) J − ∂ µ [ Ψ † K Γ µ Ψ )] (cid:9) +Tr (cid:8) J − [ Ψ † ∂ µ ( K Γ µ ) Ψ )] (cid:9) , Tr (cid:8) Ψ † K (Γ µ Ψ A µ ) J − + J − A † µ Ψ † ( K Γ µ ) † Ψ (cid:9) = Tr (cid:8)(cid:2) A µ J − + J − A † µ (cid:3) Ψ † ( K Γ µ ) Ψ (cid:9) = 0 , Tr (cid:8) Ψ † KM Ψ J − + J − Ψ † M † K † Ψ (cid:9) = Tr (cid:8) J − Ψ † KM Ψ + J − Ψ † M † K † Ψ (cid:9) == Tr (cid:8) J − (cid:2) Ψ † ( KM + M † K † ) Ψ (cid:3)(cid:9) . Ultimately we find L ψ − L ψ = ∂ µ Tr (cid:8) J − [ Ψ † K Γ µ Ψ )] (cid:9) = ∂ µ Tr (cid:8) [ Ψ † K Γ µ Ψ )] J − (cid:9) . (2.23)The Euler-Lagrange equations are K (cid:0) Γ µ ∂ µ Ψ + Γ µ Ψ A µ + M Ψ (cid:1) J − = 0 , (2.24)from which K and J − can be cancelled. c) The Lagrangian density L ψ = Tr (cid:8) [ ∂ µ Ψ ] † Θ µν [ ∂ ν Ψ ] + Ψ † R Ψ (cid:9) , (2.25)with Θ µν , R : R N → C r × r and [Θ µν ] † = Θ νµ , R † = R , is R -valued. It leads to the 2ndorder equation X µ,ν ∂∂x µ Θ µν ∂∂x ν Ψ − R Ψ = 0 . (2.26) d. The Lagrangian density for functions Ψ = col[ ψ ψ ] : R N +1 → C , L ψ = Tr h Ψ † ( i ∂ t Ψ + ∆ Ψ + V Ψ ) i , with x V ( x ) ∈ C × , V † = V , (2.27)leads to a R -valued Lagrangian functional L . Indeed L ψ − L ψ = i ∂ t Tr h Ψ † Ψ i + ∂ x Tr h Ψ † ( ∂ x ) Ψ − ( ∂ x Ψ ) † Ψ i + . . . + ∂ x N Tr h Ψ † ( ∂ x N ) Ψ − ( ∂ x N Ψ ) † Ψ i . The L ψ of (2.27) leads to the Schrödinger equation for a particle with spin .12 Free Gauge Fields
The ’field variables’ to be considered in this section are smooth functions A : R N → C c × c × · · · × C c × c | {z } N times : x
7→ A ( x ) = col[ A ( x ) , . . . , A µ ( x ) , . . . , A N ( x )] , (3.1)with A µ ( x ) ∈ g , with g ⊂ C c × c some fixed real Lie algebra. This means that g is a R -linear subspace in C c × c which is not necessarily a C -linear subspace. On g we imposethe usual ’commutator’-Lie product { A µ , A ν } = (cid:0) A µ A ν − A ν A µ (cid:1) . Important examples are matrix Lie Algebras of type g J = { X ∈ C r × r (cid:12)(cid:12) X † J + J X = 0 } , with fixed invertible J ∈ C r × r . Note that g J is always a R -linear subspace in C r × r , but not necessarily C -linear.However: { J − = J † } ⇒ { X ∈ g J ⇒ X † ∈ g J } .Next, by P g : C c × c → g , we denote the real orthogonal projection with respect to the real inner product X, Y Re Tr[ X † Y ] . Remarks 3.1
Consider C c × c as a real vector space with standard real inner product X, Y Re Tr[ X † Y ] .By P g : C c × c → g , we denote the real orthogonal projection with respect this inner product. • The Hermitean conjugation map X X † is R -linear symmetric and orthogonal. • If ∀ X ∈ g : X † ∈ g , in short g † = g , it follows that ∀ X ∈ C c × c : P g ( X † ) = ( P g X ) † . • For fixed
K, L ∈ C c × c the mapping X KX † L is R -linear. Its R -adjoint is Y LY † K . • For any fixed invertble J ∈ C c × c the mapping Q J : C c × c → C c × c : X Q J X = 12 ( X − J − X † J ) , (3.2) is a R -linear mapping which reduces to the identity map when restricted to g J . • Q J is a R -linear projection on g J iff J = J † . • Q J is a R -linear orthogonal projection on g J if J = J − = J † .In this special case Q J = P g , with g = g J . In physics textbooks one often denotes i A µ , instead of A µ , cf. [DM]. For resemblance with Electro-magnetism, I suppose. Because of u (1) = i R ? To this author the factor i is not convenient in all othercases. If we modify the standard real inner product on C c × c to X, Y Re Tr[ X † J Y ] , theprojection Q J is orthogonal iff J = J † . Proof • Re Tr[( X † ) † Y ] = Re Tr[ XY ] = Re Tr[ X † ( Y † )] . Also Re Tr[( X † ) † ( Y † )] = Re Tr[( X ) † ( Y )] . • Since g is supposed to be an invariant subspace for X X † and the latter is symmetric,also g ⊥ is invariant. • Re Tr[( KX † L ) † Y ] = Re Tr[ KX † LY † ] = Re Tr[ X † ( LY † K )] . • For X ∈ g holds ( I − Q J ) X = 0 , iff X ∈ g . • Q J = Q J iff J = J † . •• Re Tr[( X − J − X † J ) † J Y ] = Re Tr[ X † J Y ] − Re Tr[ X † J ( J − Y † J † J − )] .The 2nd term equals − Re Tr[ X † J ( J − Y † J ] , for all X, Y , iff J = J † . (cid:4) Associated with A , cf. (3.1), we introduce covariant-type partial derivatives ∇ Aµ , ≤ µ ≤ N of functions U ∈ C ∞ ( R N : C c × c ) by ∇ Aµ U = ∂ µ U − { A µ , U } = ∂ µ U − ad A µ U . (3.3)One has the Leibniz-type rules ∇ Aµ ( U V ) = ( ∇ Aµ U ) V + U ( ∇ Aµ V ) , Tr (cid:2) U ( ∇ Aµ V ) (cid:3) = ∂ µ Tr (cid:2) U V (cid:3) − Tr (cid:2) ( ∇ Aµ U ) V (cid:3) . (3.4)Note that if U ∈ C ∞ ( R N : g ) then also ∇ Aµ U ∈ C ∞ ( R N : g ) .Next, as in section 1, for given A µ , A ν ∈ C ∞ ( R N : g ) , ≤ µ, ν ≤ N , define F µν = ∂ µ A ν − ∂ ν A µ − { A µ , A ν } ∈ C ∞ ( R N : g ) , (3.5)to which Theorem 1.2 applies.For the construction of a R -valued Lagrangian density G A for the Gauge field(s) A we againemploy a proto Lagrangian G , which is now an analytic function of N ( N − complex-matrix variables and just smooth in N real variables: G : C c × c × · · · × C c × c | {z } N ( N − times × C c × c × · · · × C c × c | {z } N ( N − times × R N → C . (3.6)The 1st set of entries to this function is labeled by the ordered pairs ( µν ) , ≤ µ < ν ≤ N .The 2nd set of entries is labelled by the ordered triple ( θρ⋆ ) , ≤ θ < ρ ≤ N . We denote { . . . , P µν , . . . ; . . . , Q θρ⋆ , . . . ; x } 7→ G ( . . . P µν , . . . ; . . . Q θρ⋆ , . . . ; x ) ∈ C , with ≤ µ < ν ≤ N and ≤ θ < ρ ≤ N . The 3 bunches of variables get theircorresponding partial derivatives denoted by, respectively, cf. (2.4), G ( µν ) ( . . . , P θρ , . . . ; . . . , Q θρ⋆ , . . . ; x ) , G ( θρ⋆ ) ( . . . , P θρ , . . . ; . . . , Q θρ⋆ , . . . ; x ) , G ( ∇ ) . g be fixed. On G we put the condition, take Q θρ⋆ = P † θρ , ∀ { P µν } ≤ µ<ν ≤ N ⊂ g ∀ x ∈ R N : G ( . . . , P µν , . . . ; . . . , P † θρ , . . . ; x ) ∈ R . (3.7)The Lagrangian density we want to consider is found by replacing P µν → F µν , Q θρ⋆ → F † θρ , x G A ( x ) = G ( . . . , F µν ( x ) , . . . ; . . . , F † θρ ( x ) , . . . ; x ) ∈ R . (3.8)Note that if g = g J , for some fixed J ∈ C c × c , we have F † θρ = − J F θρ J − , θ < ρ .As in the previous section, a corresponding useful notation is x G ( µν ) A ( x ) = G ( µν ) ( . . . , F µν ( x ) , . . . ; . . . , F † θρ ( x ) , . . . ; x ) ∈ C c × c . (3.9)The Lagrangian density G A depends on the field variables x
7→ A µ ( x ) , ≤ µ ≤ N , andtheir derivatives. All being functions in a vectorspace over R . In the important specialcase g = g J the hermitean conjugate notation of the field variables A µ need not even occur.Finally, note that, because of (2.11) and (3.8), we have G ( θρ⋆ ) A ( x ) = ( G ( θρ ) A ) † ( x ) , ≤ θ < ρ ≤ N . (3.10)
Notation 3.2
In order to visually simplify the formulae to come, it is useful to extendthe set of functions G ( µν ) A , cf.(3.9), to ’full’ labels ≤ µ, ν ≤ N in the following way, ˆ G ( µν ) A = G ( µν ) A if ≤ µ < ν ≤ N , as before , if µ = ν , − G ( νµ ) A if ≤ ν < µ ≤ N . (3.11)
Theorem 3.3
Fix a matrix Lie algebra g ⊂ C c × c . Consider the Lagrangian density G A of (3.8). A. The Euler-Lagrange equations for the free gauge fields A µ , ≤ µ ≤ N , with values inthe Lie algebra g ⊂ C c × c , read N X µ =1 P g (cid:16)(cid:16) ∇ Aµ (cid:0) [ P g ˆ G ( µκ⋆ ) A ] † (cid:1)(cid:17) † (cid:17) = 0 , ≤ κ ≤ N , (3.12) with ∇ Aµ as in (3.3). B. In the special case g † = g the Euler-Lagrange equations simplify to N X µ =1 (cid:16) ∇ Aµ P g ˆ G ( µκ ) A (cid:17) = 0 , ≤ κ ≤ N . (3.13) C. If we take g = g J , with J = J † = J − , the latter becomes N X µ =1 ∇ Aµ (cid:16) Q J [ ˆ G ( µκ ) A ] (cid:17) = 0 , ≤ κ ≤ N , (3.14) where Q J Z = Z − J Z † J , Z ∈ C c × c . roofA. In order to calculate the (directional) derivatives of the Lagrangian functional G = R G A d x with respect to the free gauge fields A κ , ≤ κ ≤ N , we first expand a perturbationof x
7→ F µν ( x ) by substitution of the gauge fields x
7→ A µ ( x ) + εδ µκ H ( x ) , ε ∈ R , F µν ; ε,κ = h ∂ µ ( A ν + εδ νκ H ) − ∂ ν ( A µ + εδ µκ H ) − { A µ + εδ µκ H , A ν + εδ νκ H } i == h ∂ µ A ν − ∂ ν A µ − { A µ , A ν } i + ε δ νκ h ∂ µ H − { A µ , H } i − ε δ µκ h ∂ ν H − { A ν , H } i == F µν + εδ νκ ∇ Aµ H − εδ µκ ∇ Aν H . Consider the expansion G ( . . . , F µν ; ε,κ , . . . ; . . . , F † θρ ; ε,κ , . . . ; x ) − G ( . . . , F µν , . . . ; . . . , F † θρ , . . . ; x ) == ε X ≤ µ<ν ≤ N Tr h [ G ( µν ) A ][ δ νκ ∇ Aµ H − δ µκ ∇ Aν H ] ++ ε X ≤ θ<ρ ≤ N Tr h [ G ( θρ⋆ ) A ][ δ ρκ ∇ Aθ H − δ θκ ∇ Aρ H κ ] † i + O ( ε ) == ε N X µ, ν =1 Tr h [ ˆ G ( µν ) A ][ δ νκ ∇ Aµ H − δ µκ ∇ Aν H ] ++ ε N X θ, ρ =1 Tr h [ ˆ G ( θρ⋆ ) A ][ δ ρκ ∇ Aθ H − δ θκ ∇ Aρ H ] † i + O ( ε ) == ε N X µ =1 Tr h [ ˆ G ( µκ ) A ][ ∇ Aµ H i − ε N X ν =1 Tr h [ ˆ G ( κν ) A ][ ∇ Aν H ] i ++ ε N X θ =1 Tr h [ ˆ G ( θκ⋆ ) A ][ ∇ Aθ H i − ε N X ρ =1 Tr h [ ˆ G ( κρ⋆ ) A ][ ∇ Aρ H ] † i + O ( ε ) == ε N X µ =1 Tr h [ ˆ G ( µκ ) A ][ ∇ Aµ H i + ε N X µ =1 Tr h [ ˆ G ( µκ⋆ ) A ][ ∇ Aµ H ] † i + O ( ε ) == 2 ε Re N X µ =1 Tr h [ ˆ G ( µκ⋆ ) A ] † [ ∇ Aµ H ] i + O ( ε ) = 2 ε Re N X µ =1 Tr h [ P g ˆ G ( µκ⋆ ) A ] † [ ∇ Aµ H ] i + O ( ε ) == − ε Re N X µ =1 Tr h ∇ Aµ (cid:0) [ P g ˆ G ( µκ⋆ ) A ] † (cid:1) H i + N X µ =1 ∂ µ ( . . . ) + O ( ε ) = − ε Re N X µ =1 Tr h (cid:16) P g (cid:16)(cid:16) ∇ Aµ (cid:0) [ P g ˆ G ( µκ⋆ ) A ] † (cid:1)(cid:17) † (cid:17)(cid:17) † H i + N X µ =1 ∂ µ ( . . . ) + O ( ε ) . (3.15)In this derivation we used, respectively, the antisymmetry µ ↔ ν of [ ˆ G ( µν ) A ] and [ δ νκ ∇ Aµ H − δ µκ ∇ Aν H ] , the Leibniz rule(3.4), the fact that Re Tr h(cid:0) . . . (cid:1) † H i expresses the real innerproduct on C c × c and P g the real orthogonal projection on g .Also properties like Tr[ AB ] = Tr[ BA ] , Tr[ A { B , C } ] = Tr[ { A , B } C ] play a crucial role.The result now follows by the usual variational practices. B. If g † = g the real linear mappings { . } † and P g commute, which greatly simplifies theresult of A. C. Use Remarks 3.1. (cid:4)
Example 3.4A.
For convenience we restrict to Lie-algebras with property g † = g . We will considergeneral Lagrangians which are (real) quadratic in F µν . Here, in our summation expressions,we write µ < ν instead of ≤ µ < ν ≤ N . Start from the proto Lagrangian G = X µ<ν , θ<ρ h ( µν )( θρ ) Tr[ P µν Q θρ⋆ ] with h ( µν )( θρ ) = h ( θρ )( µν ) ∈ C . (3.16)Note X µ<ν,θ<ρ h ( µν )( θρ ) Tr[ P µν P † θρ ] ∈ R . For the derivatives of G we find, G ( µν ) ( . . . , P µν , . . . ; . . . , Q θρ⋆ , . . . ) = X α<β h ( µν )( αβ ) Q αβ⋆ G ( θρ⋆ ) ( . . . , P µν , . . . ; . . . , Q θρ⋆ , . . . ) = X α<β h ( αβ )( θρ ) P αβ If we take Q θρ⋆ = P † θρ , one easily checks (3.8), G ( µν ) † ( . . . , P µν , . . . ; . . . , P † θρ , . . . ) = X α<β h ( µν )( αβ ) P αβ = X α<β h ( αβ )( µν ) P αβ = G ( µν⋆ ) . The Lagrangian density G A = X µ<ν, θ<ρ h ( µν )( θρ ) Tr[ F µν F † θρ ] , (3.17)can now be put in (3.13) to find the Euler-Lagrange equations. Note however, that P g cannot be put ’through’ the h ( µν )( θρ ) if those are non-real numbers!17o, let us restrict to g † = g ànd h ( µν )( θρ ) ∈ R . Anti-symmetrize h ( µν )( θρ ) to full labels: ˆ h ( µν )( θρ ) = h ( µν )( θρ ) if µ < ν , θ < ρ or µ > ν , θ > ρ if µ = ν and/or θ = ρ − h ( νµ )( θρ ) if µ > ν , θ < ρ − h ( µν )( ρθ ) if µ < ν , θ > ρ In this special case ˆ G ( µν ) A = 12 N X α,β =1 ˆ h ( µν )( αβ ) F † αβ , and, since F † αβ ∈ g , the E-L-equations (3.13) become N X α,β =1 N X µ =1 ˆ h ( µκ )( αβ ) (cid:16) ∂ µ F † αβ − { A µ , F † αβ } (cid:17) = 0 , ≤ κ ≤ N . (3.18) B. For gauge fields on Minkowski space, with coordinates x , x , x , x andmetric [ g µν ] = diag(1 , − , − , − , one usually takes, cf. [DM], h ( µν )( αβ ) = g µα g νβ = ( − δ µ δ µα ( − δ ν δ νβ = ( − δ µ + δ ν δ µα δ νβ . Hence ˆ h ( µκ )( αβ ) = sgn( κ − µ ) sgn( β − α ) ( − δ µ + δ κ δ µα δ κβ . In this special case the Lagrangian density (3.17) reads G A = X ≤ µ<ν ≤ ( − δ µ + δ ν Tr (cid:2) F µν F † µν (cid:3) . (3.19)The corresponding Euler-Lagrange equations are X µ =0 ( − δ µ + δ κ ∇ Aµ F † µκ = 0 , ≤ κ ≤ . (3.20)For dim g = 1 the term ad A µ F † µκ vanishes. This simplification, viz. ∇ Aµ = ∂ µ , leadsto standard electromagnetism in Minkowski space. Indeed, if we put A † = − Φ and col[ A † , A † , A † ] = A , then (3.20) turns into Maxwell’s equations ’in potential form’ ∂∂t div A + ∆Φ = 0 ∂ ∂t A − ∆ A + grad (cid:0) ∂∂t Φ + div A (cid:1) = 0 (3.21)18f the pair A, B satisfies (3.21), then the pair E = − ∂ A∂t − gradΦ , B = rot A , satisfies theclassical Maxwell equations.Finally, imposing the ’Lorenz-Gauge’ ∂∂t Φ + div A = 0 , we find the usual wave equations ∂ t Φ − ∆Φ = 0 , ∂ t A − ∆ A = 0 . For more details see Appendix B. ’Infinitesimal symmetries’ of the Lagrangian density L lead to local conservation laws forthe solutions of the Euler Lagrange equations. So we are told by Emmy Noether’s famoustheorem. First we have a short look at the needed concepts as formulated within ourspecial (simple) context. Definition 4.1 A Conservation Law or Noether Flux is a vectorfield on R N , with com-ponents V µψ , ≤ µ ≤ N , which arise from a set of functions of Proto-Lagrangian type, V µ , ≤ µ ≤ N , cf. (2.13), such that for all solutions Ψ of the Euler Lagrangian system,cf. Th 2.4, we have N X µ =1 ∂∂x µ V µψ ( x ) = 0 , where V µψ ( x ) = V µ ( Ψ ( x ) , Ψ † ( x ) , Ψ ,µ ( x ) , Ψ † ,µ ( x ) , x ) . (4.1)A conservation law can be named ’trivial’ for several reasons: It may happen that for all solutions Ψ the fluxes V µψ = 0 . Another reason for triviality occurs if for all functions Ψ , whether they are solutions or not, the identity (4.1) is satisfied. For example if thecomponents V µψ arise from the curl of an arbitrary vector field depending on Ψ .Two types of symmetries will be considered here: ’Internal symmetries’ and ’Externalsymmetries’. They can be formulated in terms of the proto-Lagrangian only.External symmetries regard transformations of the spatial variables x . We restrict to affinetransforms . Definition 4.2 ( Internal symmetries )A set of linear mappings K , L λµ : C r × c → C r × c , ≤ λ, µ ≤ N , is said to generate aninternal (local) symmetry of the proto-Lagrangian L if for all P , Q µ ∈ C r × c , all x ∈ R N ,and s ∈ R , | s | small, one has L ( e s K P; ( e s K P) † ; . . . e s L λµ Q λ . . . ; . . . ( e s L λµ Q λ ) † . . . ; x ) == L (P; P † ; . . . Q µ . . . ; . . . Q † µ . . . ; x ) + O ( s ) , (4.2)In many cases the K , L λµ are realized by left and/or right multiplication with some fixedmatrices in C r × r or C c × c . 19any times there is a special type of internal symmetry which is related to a linear mapping A : R N → R N in the ’outside world’, L (P; P † ; . . . ( e sA ) λµ Q λ . . . ; . . . (( e sA ) λµ Q λ ) † . . . ; x ) == L (P; P † ; . . . Q µ . . . ; . . . Q † µ . . . ; x ) + O ( s ) , (4.3) Definition 4.3 ( External symmetries )The affine mapping x
7→ − sa + e sA x on R N , where a ∈ R N and A : R N → R N , a linearmapping, is said to generate an external (local) symmetry of the proto-Lagrangian L iffor all P , Q µ ∈ C r × c , all x ∈ R N , and s ∈ R , | s | small, one has L (P; P † ; . . . Q µ . . . ; . . . (Q µ ) † . . . ; − sa + e sA x ) == L (P; P † ; . . . Q µ . . . ; . . . Q † µ . . . ; x ) + O ( s ) . (4.4) Remarks 4.4 • The order constant in O ( s ) may depend on all independent variables of L . • If in (4.2)-(4.4) exponents like e s K are replaced by I + s K we get equivalent conditi-ons. However in many practical applications the terms O ( s ) are identically zero ifexponentials are used. • Local symmetry (4.4) implies L ( ∇ ) (P; P † ; . . . Q µ . . . ; . . . Q † µ . . . ; x ) · ( Ax − a ) = 0 . We now first consider two types of conservation laws in connection with affine transforma-tions in space.For any vector a ∈ R N we define the Translation operator T a by T a Ψ ( x ) = Ψ ( x − a ) . For any matrix A ∈ R N × N we define the dilation operator R A by R A Ψ ( x ) = Ψ ( e A x ) . Theorem 4.5
Suppose that, for some
K : C r × c → C r × c and some a ∈ R N , the proto-Lagrangian L has internal local symmetry (4.2) with L λµ = δ λµ K and external local symmetry (4.4) with A = O . Then for any solution Ψ of the Euler-Lagrange system one has the conservationlaw N X µ =1 ∂∂x µ n Tr h [ L ( µ ) ψ ] · (K Ψ − a λ ∂ λ Ψ ) + [ L ( µ⋆ ) ψ ] · (K Ψ − a λ ∂ λ Ψ ) † i + a µ L ψ o = 0 . (4.5)20 roof: By ∼ = we mean equality up to a term O ( s ) . We study L (cid:0) e s K T sa Ψ , T sa Ψ † e s K † , ∂ µ [ e sK T sa Ψ ] , ∂ µ [ T sa Ψ † e sK † ] , x − sa (cid:1) . With our conditions it can be written L ( e s K Ψ ( x − sa ); ( e s K Ψ ( x − sa )) † ; . . . ∂ µ e s K Ψ ( x − sa ) . . . ; . . . ∂ µ ( e s K Ψ ( x − sa )) † . . . ; x − sa ) ∼ = ∼ = L ( Ψ ( x − sa ); Ψ ( x − sa ) † ; . . . Ψ ,µ ( x − sa ) . . . ; . . . Ψ ,µ ( x − sa ) † . . . ; x − sa ) == L ψ ( x − sa ) = ( T sa L ψ )( x ) . (4.6)Differentiate the first line of this at s = 0 and use L ( ∇ ) · a = 0 , Tr (cid:8) [ L ( o ) ψ ](K Ψ − a λ ∂ λ Ψ ) + [ L ( o⋆ ) ψ ]( Ψ † K † − a λ ∂ λ Ψ † )++ [ L ( µ ) ψ ](K ∂ µ Ψ − a λ ∂ λ ∂ µ Ψ ) + [ L ( µ⋆ ) ψ ]( ∂ µ Ψ † K † − a λ ∂ λ ∂ µ Ψ † ) (cid:9) . (4.7)If Ψ is a solution we use (2.16) and replace [ L ( o ) ψ ] by ∂∂x µ [ L ( µ ) ψ ] , etc. Now (4.7) can bewritten as a divergence, which constitutes the left hand side of (4.5), apart from the lastterm inside { } . Together with the derivative a λ ∂ λ L ψ = ∂ µ ( a µ L ψ ) at s = 0 of the finalline of (4.6) we arrive at the wanted conserved current (4.5). (cid:4) Example 4.6
Let Γ µ and M be constant complex matrices with Γ µ † = Γ µ and M = − M † .Then the Lagrangian density L ψ = Tr (cid:8) i Ψ † Γ µ ∂ µ Ψ + Ψ † M Ψ (cid:9) , (4.8)for Ψ : R N → C r × c satisfies the condition of Theorem 4.1 for K = O and all a ∈ R N .The conservation law reads ∂∂x µ Tr (cid:8) − a λ Ψ † Γ µ ∂ λ Ψ + a µ Ψ † Γ λ ∂ λ Ψ + a µ Ψ † M Ψ (cid:9) = ∂∂x µ Tr (cid:8) − a λ Ψ † Γ µ ∂ λ Ψ (cid:9) = 0 . (4.9)This can be checked directly for solutions of the PDE: Γ µ ∂ µ Ψ + M Ψ = 0 . Observe thatin this special case L ψ = 0 for solutions.Also the Lagrangian of Example (2.5b), with constant matrices K, M, Γ µ , A µ leads toconservation laws of this type. Theorem 4.7
Suppose that, for some
K : C r × c → C r × c and some A ∈ R N × N with Tr A = 0 , the proto-Lagrangian L has internal local symmetry (4.2) with L λµ = K + [ A ] λµ I and external localsymmetry (4.4) with a = 0 . Then for any solution Ψ of the Euler-Lagrange system onehas the conservation law N X µ =1 ∂∂x µ n Tr h [ L ( µ ) ψ ](K Ψ ( x ) + A αβ x β Ψ ,α ( x )) ++ [ L ( µ⋆ ) ψ ](K Ψ ( x ) + A αβ x β Ψ ,α ( x )) † i − A µβ x β L ψ o = 0 . (4.10)21 roof: We study L (cid:0) e s K R sA Ψ ; R sA Ψ † e s K † ; . . . ∂ µ [ e s K R sA Ψ ] . . . ; . . . ∂ µ [ R sA Ψ † e s K † ] . . . ; e sA x (cid:1) . With our conditions it can be written, L ( Ψ ( e sA x ); Ψ ( e sA x ) † ; . . . ∂ µ Ψ ( e sA x ) . . . ; . . . ∂ µ Ψ ( e sA x ) † . . . ; e sA x ) ∼ = ∼ = L ( Ψ ( e sA x ); Ψ ( e sA x ) † ; . . . ( e sA ) λµ Ψ ,λ ( e sA x ) . . . ; . . . ( e sA ) λµ Ψ ,λ ( e sA x ) † . . . ; e sA x ) ∼ = ∼ = L ( Ψ ( e sA x ); Ψ ( e sA x ) † ; . . . Ψ ,µ ( e sA x ) . . . ; . . . Ψ ,µ ( e sA x ) † . . . ; e sA x ) ∼ = ∼ = L ( Ψ ( e sA x ); Ψ ( e sA x ) † ; . . . Ψ ,µ ( e sA x ) . . . ; . . . Ψ ,µ ( e sA x ) † . . . ; e sA x ) == L ψ ( e sA x ) = ( R sA L ψ )( x ) . (4.11)Differentiate the first line of this at s = 0 and use L ( ∇ ) · Ax = 0 : Tr (cid:8) [ L ( o ) ψ ](K Ψ ( x )+ A αβ x β Ψ ,α ( x ))+[ L ( µ ) ψ ] ∂ µ (K Ψ ( x )+ A αβ x β Ψ ,α ( x )) ++ [ L ( o⋆ ) ψ ](K Ψ ( x ) + A αβ x β Ψ ,α ( x )) † + [ L ( µ⋆ ) ψ ] ∂ µ (K Ψ ( x ) + A αβ x β Ψ ,α ( x )) † (cid:9) . (4.12)If Ψ is a solution we use (2.16) and replace [ L ( o ) ψ ] by ∂∂x µ [ L ( µ ) ψ ] , etc. Now (4.12) canbe written as a divergence, which constitutes the left hand side of (4.10), apart from thelast term between { } . Together with the derivative at s = 0 of the final line in (4.11): A µβ ∂ µ L ψ = ∂ µ ( A µβ x β L ψ ) , use Tr A = 0 , we arrive at the conserved current (4.10). (cid:4) Next we deal with internal symmetries only. They play a crucial role in Gauge theories.A simple case first.
Theorem 4.8
Suppose that, for some linear
K : C r × c → C r × c the proto-Lagrangian L satisfies (4.2) with L λµ = δ λµ K . Then for any solution Ψ of the Euler-Lagrange system one has the conservationlaw N X µ =1 ∂∂x µ Tr (cid:8) [ L ( µ ) ψ ]K Ψ + [ L ( µ⋆ ) ψ ](K Ψ ) † (cid:9) = 0 , (4.13) Proof:
Calculate the derivative ∂∂s L (cid:0) e s K Ψ , ( e s K Ψ ) † , ∂ µ [ e s K Ψ ] , ∂ µ [ e s K Ψ ] † , x (cid:1) , at s = 0 . With the notation of (2.5) one finds Tr (cid:8) [ L ( o ) ψ ][K Ψ ] + [ L ( o⋆ ) ψ ][K Ψ ] † + [ L ( µ ) ψ ][K Ψ ,µ ] + [ L ( µ⋆ ) ψ ][K Ψ ,µ ] † (cid:9) = 0 . Ψ happens to be a solution of the Lagrangian system, then with (2.16) this becomes Tr (cid:8) [ ∂∂x µ L ( µ ) ψ ][K Ψ ] + [ ∂∂x µ L ( µ⋆ ) ψ ][K Ψ ] † + [ L ( µ ) ψ ][K Ψ ] , µ + [ L ( µ⋆ ) ψ ][K Ψ ] † , µ (cid:9) = 0 , which leads to the wanted ’conserved current’, since K is supposedly constant. (cid:4) In gauge applications K is often realized by a right multiplication by some A ∈ C c × c . Insuch cases K Ψ in (4.13) should be replaced by Ψ A .All previous considerations can be applied to matrix gauge fields as well if we replace Ψ by A = col[ . . . , A µ , . . . ] . Some subtleties occur however because the range of the functions A µ is not the whole of C c × c but some real linear subspace g of it. See Appendix A for moredetails.This section is concluded with conservation laws for non-commutative free gauge fieldswhich come from the special Lagrangian density (3.8). Theorem 4.9
Consider the proto-Lagrangian G of (3.6) with property (3.7) and Lagrange density asdenoted in (3.8). For convenience restrict to g = g † only. a. Suppose G ( ∇ ) A · a = 0 , for some a ∈ R N then we have the conservation law N X µ =1 ∂∂x µ (cid:16) N X κ =1 Re Tr h P g ˆ G ( µκ ) A : ( a ·∇ ) A κ i − a µ G A (cid:17) = 0 . (4.14) b. If for some S = [ S λµ ] ∈ R N × N , with Tr S = 0 , the assumptions G ( ∇ ) A · Sx = 0 and Re N X µ, ν =1 Tr h ˆ G ( µν ) A : N X α =1 S αµ ∂ α A ν i = 0 , (4.15) hold, then we have the conservation law N X µ =1 ∂∂x µ (cid:16) N X κ =1 h P g ˆ G ( µκ ) A ( Sx · ∇ ) A κ i − ( Sx · e µ ) G A (cid:17) = 0 . (4.16) Proofa.
Start from dd s G ( . . . , F µν ( x − sa ) , . . . ; . . . , F † θρ ( x − sa ) , . . . ; x − sa ) (cid:12)(cid:12)(cid:12) s =0 = dd s G A ( x − sa ) (cid:12)(cid:12)(cid:12) s =0 . Calculate the left hand side with the chain rule and use the assumptions − X µ<ν Tr h G ( µν ) A : ( a · ∇ ) F µν i − X µ<ν Tr h G ( µν⋆ ) A : ( a · ∇ ) F † µν i − a · G ∇ A = − X µ<ν Tr h G ( µν ) A : ( a · ∇ ) F µν i . (4.17)With ( a · ∇ ) F µν = ∂ µ ( a · ∇A ν ) − ∂ ν ( a · ∇A µ ) − { A µ , a · ∇A ν } + { A ν , a · ∇A µ } , and the antisymmetries µ ↔ ν , the expression (4.17) becomes, (mind the hat ˆ ), − Re N X µ,ν =1 Tr h ˆ G ( µν ) A : ∂ µ ( a · ∇A ν ) − { A µ , a · ∇A ν } i = − Re N X µ,ν =1 ∂∂x µ Tr h ˆ G ( µν ) A : ( a ·∇A ν ) i + Re N X µ,ν =1 Tr h ∂ µ ˆ G ( µν ) A : ( a ·∇A ν ) + ˆ G ( µν ) A : { A µ , a ·∇A ν } i . The 2nd term is equal to Re N X ν =1 N X µ =1 Tr h ∇ Aµ P g ˆ G ( µν ) A : ( a · ∇A ν ) i = 0 , because of the E-L-equations (3.13).The right hand side of the 1st formula of this proof equals − ∂ µ ( a µ L A ) . Hence (4.14). b. Start from dd s G ( . . . , F µν ( e sS x ) , . . . ; . . . , F † θρ ( e sS x ) , . . . ; e sS x ) (cid:12)(cid:12)(cid:12) s =0 = dd s G A ( e sS x ) (cid:12)(cid:12)(cid:12) s =0 . Calculate the left hand side with the chain rule and use G ( ∇ ) A · Sx = 0 , X µ<ν Tr h G ( µν ) A : ( Sx · ∇ ) F µν i == Re N X µ, ν =1 Tr h ˆ G ( µν ) A : ∂ µ (cid:0) ( Sx · ∇ ) A ν (cid:1) − { A µ , ( Sx · ∇ ) A ν } − S αµ ∂ α A ν i . Because of the assumption the very final contribution vanishes. Then we proceed as inpart a . (cid:4) Note
The orthogonality condition (4.15) is inspired by combining Thm 4.7 with AppendixA. Indeed, another way to obtain the preceding Theorem is to rewrite Thms 4.5, 4.7 interms of A with the aid of the table in Appendix A. Theorem 4.10
Consider the proto-Lagrangian G of (3.6) with property (3.7) and Lagrange density asdenoted in (3.8). For convenience consider g = g † only. Suppose G satisfies G ( . . . , e s B P µν e − s B , . . . ; . . . , e − s B † P † θρ e s B † , . . . ; x ) = G ( . . . , P µν , . . . ; . . . , P † θρ , . . . ; x ) , (4.18)24 or all P µν ∈ g ⊂ C c × c , ≤ µ < ν ≤ N , some fixed B ∈ g and (small) s ∈ R .Then, for any solution x . . . A µ ( x ) . . . of the Lagrangian system of Theorem 3.3 one hasthe conservation law N X µ =1 ∂∂x µ Re (cid:16) N X ν =1 Tr h [ ˆ G ( µν ) A ] : { B , A ν } i(cid:17) = 0 . (4.19) Proof
In (4.18) replace P µν → F µν and Q θρ → F † θρ and put the derivative to s equal to at s = 0 , X ≤ µ<ν ≤ N Tr h [ G ( µν ) A ] : ( BF µν − F µν B )] i + X ≤ θ<ρ ≤ N Tr h [ G ( θρ⋆ ) A ] : ( −B † F † θρ + F † θρ B † )] i = 0 . (4.20)Due to the anti-symmetry in µ ↔ ν of BF µν − F µν B = ∂ µ { B , A ν } − ∂ ν { B , A µ } − { B , { A µ , A ν }} , applying convention (3.11), together with G ( µν ⋆ ) A = [ G ( µν ) A ] † , the 1st term of (4.20) equalsthe Re -part of N X µ =1 N X ν =1 Tr h [ ˆ G ( µν ) A ] : ( BF µν − F µν B ) i == N X µ =1 N X ν =1 ∂∂x µ Tr h [ ˆ G ( µν ) A ] { B , A ν } i − N X ν =1 N X µ =1 ∂∂x ν Tr h [ ˆ G ( µν ) A ] { B , A µ } i + − N X ν =1 N X µ =1 Tr h [ ∂ µ ˆ G ( µν ) A ] { B , A ν } i + N X µ =1 N X ν =1 Tr h [ ∂ ν ˆ G ( µν ) A ] { B , A µ } i + − N X µ =1 N X ν =1 Tr h [ ˆ G ( µν ) A ] { B , { A µ , A ν }} i . (4.21)On the 2nd line we apply the E-L-equations (3.13) together with ∂ ν ˆ G ( µν ) A = − ∂ ν ˆ G ( νµ ) A .This together with the 3rd line leads to − N X ν =1 N X µ =1 Tr h { A µ , ˆ G ( µν ) A }{ B , A ν } i + N X µ =1 N X ν =1 Tr h { A ν , ˆ G ( µν ) A }{ B , A µ } i + − N X µ =1 N X ν =1 Tr h [ ˆ G ( µν ) A ] { B , { A µ , A ν }} i . These 3 terms add up to because for each pair µ, ν separately we can apply the identity − Tr h { M , G } : { B , N } i + Tr h { N , G } : { B , M } i = Tr h G : { B , { M , N }} i , (4.22)25or matrices G , B , M , N ∈ C r × r .(Of course the two terms on the 3rd line of (4.21) are equal. But then, using that equality,the latter trick no longer works for each index pair µ, ν separately!)Thus we found out that (4.20) corresponds to (4.19). (cid:4) A basic ingredient for this section is a (fixed) Lie-group G ⊂ C c × c of invertible c × c -matrices.Its Lie-algebra g is a R -linear subspace of C c × c . Important examples are (subgroups of) G J , for some fixed invertible matrix J ∈ C c × c . The relevant definitions are as in section 3, G J = (cid:8) U ∈ C c × c (cid:12)(cid:12) U † J U = J (cid:9) , g J = (cid:8) A ∈ C c × c (cid:12)(cid:12) A † J + J A = 0 (cid:9) . (5.1)In the discussion to follow suitable subspaces ofthe group G loc = C ∞ ( R N : G ) and the R -linear space C ∞ ( R N : g ) will be used. It will be tacitly assumed that the behaviour at ∞ of the considered subspacesis such that our formulae make sense. The C ∞ -smoothness condition can often be relaxed.Neither of those assumptions will bother us.The group action from the right of C ∞ ( R N : G ) on C ∞ ( R N : C r × c ) is naturally defined by C ∞ ( R N : C r × c ) × C ∞ ( R N : G ) → C ∞ ( R N : C r × c ) : ( Ψ U )( x ) = Ψ ( x ) U ( x ) . For each ≤ µ ≤ N , a group action from the right of C ∞ ( R N : G ) on C ∞ ( R N : g ) isdefined by C ∞ ( R N : g ) × C ∞ ( R N : G ) → C ∞ ( R N : g ) : ( A µ ⊳ U )( x ) = U − ( x ) A µ ( x ) U ( x ) −U − ( x )( ∂ µ U )( x ) . In the proof of Thm 1.2 it has been shown that this action (’gauge transform’)is indeed a(inhomogeneous) group action. This means [ A µ ⊳ U ] ⊳ V = A µ ⊳ ( U V ) . (5.2)As before, for given A µ , A ν ∈ C ∞ ( R N : g ) , ≤ µ, ν ≤ N , define F µν = ∂ µ A ν − ∂ ν A µ − { A µ , A ν } ∈ C ∞ ( R N : g ) . (5.3)Then U − F µν U = ∂ µ ( A ν ⊳ U ) − ∂ ν ( A µ ⊳ U ) − { ( A µ ⊳ U ) , ( A ν ⊳ U ) } . (5.4) Theorem 5.1
Fix a matrix Lie-Group G ⊂ C c × c . Suppose a proto-Lagrangian L , cf. (2.13), to be G -invariant, i.e. ∀ U ∈ G ∀ P ∈ C r × c ∀ R ∈ C Nr × c ∀ x ∈ R N : Property (5.5) is named
Global Gauge Invariance by physicists. The conclusion of Theorem 5.1 isnamed, in physicists’ vernacular, the property of
Local Gauge Invariance . In mathematicians’ jargonhowever, the usage of ’global’, as opposed to ’local’, usually refers to a more involved (more difficult)notion. (P U ; U † P † ; R U ; U † R † ; x ) = L (P ; P † ; R ; R † ; x ) (5.5) Then, for all x ∈ R N , the statically gauge extended Lagrangian density L ψ, A ( x ) = L ( Ψ ; Ψ † ; . . . , ∂ µ Ψ + Ψ A µ , . . . ; . . . , ∂ µ Ψ † + A † µ Ψ † , . . . ; x ) , (5.6) with any Ψ ∈ C ∞ ( R N : C r × c ) , A µ ∈ C ∞ ( R N : g ) , ≤ µ ≤ N , equals the statically gauge extended
Lagrangian density L ψ U , A ⊳ U ( x ) = (5.7) = L ( Ψ U ; U † Ψ † ; . . . , ∂ µ ( Ψ U )+( Ψ U )( A µ ⊳ U ) , . . . ; . . . , ∂ µ ( Ψ U ) † +( A µ ⊳ U ) † ( Ψ U ) † , . . . ; x ) , with any U ∈ C ∞ ( R N : G ) . In (5.6),(5.7) we wrote Ψ instead of Ψ ( x ) , etc. Proof
Straightforward calculation. (cid:4)
Example 5.2
Consider the proto-Lagrangian, cf. (2.13), L (P; Q ⊤ ; R ; S ⊤ ; x ) = i Tr[Q ⊤ ( X µ Γ µ R µ + M P)] with fixed Γ µ , M ∈ C r × r and [Γ µ ] † = Γ µ , M † = − M . Put G = U ( c ) ⊂ C c × c , that is theunitary group G I , with I the identity matrix. Our proto-Lagrangian is U ( c ) -invariant i Tr[ U † P † (Γ µ R µ U + M P U )] = i Tr[P † (Γ µ R µ + M P)] , U ∈ U ( c ) , because U † = U − and the properties of Tr .Then the statically extended Lagrangian density L ψ, A ( x ) = i Tr[ Ψ † (cid:0) Γ µ ( ∂ µ Ψ + Ψ A µ ) + M Ψ (cid:1) ] , (5.8)with any Ψ ∈ C ∞ ( R N : C r × c ) , A µ ∈ C ∞ ( R N : u ( c )) , ≤ µ ≤ N , equals the statically extended Lagrangian density L ψ U , A ⊳ U ( x ) = i Tr[ U † Ψ † (cid:0) Γ µ ( ∂ µ ( Ψ U ) + Ψ U ( U − A µ U − U − ∂ µ U )) + M Ψ U (cid:1) ] , (5.9)with any U ∈ C ∞ ( R N : U ( c )) . Note that, if M is replaced by the ’nonlinearity’ i ΨΨ † , the argument still holds. (cid:4) heorem 5.3 • Suppose that the statically gauge extended Lagrange density L ψ, A , cf. (5.6) leads to an R -valued Langrangian functional L ψ, A . The E-L-equations are L ( o ) ψ,A − N X µ =1 (cid:16) ∂∂x µ [ L ( µ ) ψ,A ] − [ A µ L ( µ ) ψ,A ] (cid:17) = 0 , P g (cid:16) Ψ † [ L ( κ ) † ψ,A + L ( κ⋆ ) ψ,A ] (cid:17) = 0 , P g (cid:16) Ψ † [ L ( κ ) † ψ,A − L ( κ⋆ ) ψ,A ]i (cid:17) = 0 , ≤ κ ≤ N . (5.10)
Here P g : C c × c → C c × c denotes the R -orthogonal projection on g . • If it happens that P g ( iZ) = i P ⊥ g Z , Z ∈ C c × c , the 2nd line in (5.10) reduces to Ψ † L ( κ ) † ψ,A + ( P g − P ⊥ g ) Ψ † L ( κ⋆ ) ψ,A = 0 , ≤ κ ≤ N . (5.11) • In the important special case g = g J , with J = J † = J − , (5.11) can be written L ( κ ) ψ,A Ψ − J Ψ † L ( κ⋆ ) ψ,A J = 0 , ≤ κ ≤ N . (5.12)
Proof • The perturbed statically extended Lagrangian L ψ, A reads L ( Ψ + ε H ; Ψ † + ε ⋆ K ; . . . , ∂ µ ( Ψ + ε H ) + ( Ψ + ε H )( A µ + ε κ δ µκ H ) , . . . ;; . . . , ∂ µ ( Ψ † + ε ⋆ K ) + ( A † µ + ε κ δ µκ H † )( Ψ † + ε ⋆ K ) , . . . ; x ) The results of dd ε (cid:12)(cid:12) ε =0 , dd ε ⋆ (cid:12)(cid:12) ε ⋆ =0 , dd ε κ (cid:12)(cid:12) ε κ =0 , ≤ κ ≤ N , being put to are,for all functions H , K , H , Tr (cid:2) L ( o ) : H (cid:3) + X µ Tr (cid:2) L ( µ ) : ∂ µ H (cid:3) + X µ Tr (cid:2) L ( µ ) : H A µ (cid:3) = 0 , Tr (cid:2) L ( o⋆ ) : K (cid:3) + X µ Tr (cid:2) L ( µ⋆ ) : ∂ µ K (cid:3) + X µ Tr (cid:2) L ( µ⋆ ) : A † µ K (cid:3) = 0 , X µ Tr (cid:2) L ( µ ) : Ψ δ µκ H (cid:3) + X µ Tr (cid:2) L ( µ⋆ ) : δ µκ H † Ψ † (cid:3) = 0 , ≤ κ ≤ N .
The usual partial integration techniques applied to the first two lines lead to the E-L-equations for Ψ . Also use Theorem 2.4.From the final line we arrive at (5.10) because of the trace identity Tr h XZ + YZ † i = Re Tr h(cid:16) X † + Y (cid:17) † Z i − i Re Tr h(cid:16) X † − Yi (cid:17) † Z i . (5.13) • If for X , Y ∈ C c × c one has P g (X + Y) = 0 and P ⊥ g (X − Y) = 0 , it follows that
X + ( P g − P ⊥ g ) Y = 0 and also Y + ( P g − P ⊥ g ) X = 0 . • In this special case ( P g − P ⊥ g ) Y = − J Y † J and P g [Y † ] = [ P g Y] † . (cid:4) xamples 5.4 Note that in the E-L-equations (5.10) the A µ occur only ’algebraically’.The ∂ µ A are not involved! a. For the Lagrangian densities from examples 2.5a and 5.2 the 2nd set of E-L-equations(5.12) does not depend on A . If we choose g = g J , the 2nd line reads Ψ † Γ κ Ψ = 0 , ≤ κ ≤ N .
It means that Ψ can only take values in a cone in C r × c . If one of the Γ κ = Γ κ † is strictlypositive, the only solutions are Ψ = 0 , the trivial ones. If a nontrivial choice for Ψ ispossible it can be substituted in the 1st E-L-equation and we are left with an algebraicequation for the A κ . b. For the Lagrangian densities from example 2.5c, again with g = g J , the 2nd set ofE-L-equations becomes N X µ =1 [ ∂ µ Ψ + Ψ A µ ] † Θ µκ Ψ − J (cid:16) N X µ =1 [ Ψ † Θ κµ [ ∂ µ Ψ + Ψ A µ ] (cid:17) J = 0 , ≤ κ ≤ N , which is algebraic in the A κ . (cid:4) Finally we want to consider the dynamically gauge extended
Lagrangian density or
Gauge field extended
Lagrangian density of type L ψ, A ( x ) + G A ( x ) . Theorem 5.5
Fix a matrix Liegroup G ⊂ C c × c with Lie algebra g ⊂ C c × c and property g † = g .Fix a proto Lagrangian of type (2.13) (P; Q ⊤ ; R ; S ⊤ ; x ) L (P; Q ⊤ ; R ; S ⊤ ; x ) , leading to a R -valued Lagrangian functional L . Require the special property ∀ P ∀ R ∀ x : P g (cid:16) P † (cid:2) L ( κ ) † (P; P † ; R ; R † ; x ) − L ( κ⋆ ) (P; P † ; R ; R † ; x ) (cid:3) i (cid:17) = 0 . (5.14) Fix a second proto Lagrangian of type (3.6) and such that ∀ R µν ∈ g : G ( . . . , R µν , . . . ; . . . , R † θρ , . . . ; x ) ∈ R . Consider the dynamically extended
Lagrangian density L ψ, A ( x ) + G A ( x ) = L ( Ψ ; Ψ † ; . . . , ∂ µ Ψ + Ψ A µ , . . . ; . . . , ∂ µ Ψ † + A † µ Ψ † , . . . ; x ) ++ G ( . . . , F µν ( x ) , . . . ; . . . , F † θρ ( x ) , . . . ; x ) (5.15)29 ith any Ψ ∈ C ∞ ( R N : C r × c ) , A µ ∈ C ∞ ( R N : g ) , ≤ µ ≤ N . • The Euler-Lagrange equations are, with L ( o ) ψ,A instead of L ( o ) ψ,A ( x ) , etc., [ L ( o ) ψ,A ] − N X µ =1 (cid:16) ∂∂x µ [ L ( µ ) ψ,A ] − [ A µ L ( µ ) ψ,A ] (cid:17) = 0 , P g (cid:16) Ψ † [ L ( κ ) † ψ,A + L ( κ⋆ ) ψ,A ] (cid:17) − P Nµ =1 (cid:16) ∂ µ P g [ ˆ G ( µκ ) A ] − { A µ , P g [ ˆ G ( µκ ) A ] } (cid:17) † = 0 , ≤ κ ≤ N . (5.16)
Here P g : C c × c → C c × c denotes the R -orthogonal projection on g . • In the special case g = g J , with J = J † = J − , the 2nd line in (5.16) can be rewritten L ( κ ) ψ,A Ψ − J Ψ † L ( κ⋆ ) ψ,A J − N X µ =1 (cid:16) ∂ µ P g [ ˆ G ( µκ ) A ] − { A µ , P g [ ˆ G ( µκ ) A ] } (cid:17) = 0 , ≤ κ ≤ N . (5.17)
Proof • The perturbed gauge supplemented Lagrangian reads L ( Ψ + ε H ; Ψ † + ε ⋆ K ; . . . , ∂ µ ( Ψ + ε H )+( Ψ + ε H )( A µ + ε κ δ µκ H ) , . . . ;; . . . , ∂ µ ( Ψ † + ε ⋆ K )+( A † µ + ε κ δ µκ H † )( Ψ † + ε ⋆ K ) , . . . ; x ) ++ G ( . . . , F µν,εκ , . . . ; . . . , F † θρ,εκ , . . . ; x ) , ≤ κ ≤ N , where F µν ; ε,κ = F µν + ε κ δ νκ h ∂ µ H − { A µ , H } i − ε κ δ µκ h ∂ ν H − { A ν , H } i , The results of dd ε (cid:12)(cid:12) ε =0 , dd ε ⋆ (cid:12)(cid:12) ε ⋆ =0 dd ε κ (cid:12)(cid:12) ε κ =0 , being put to are, respectively, Tr (cid:2) L ( o ) : H (cid:3) + X µ Tr (cid:2) L ( µ ) : ∂ µ H (cid:3) + X µ Tr (cid:2) L ( µ ) : H A µ (cid:3) = 0 , Tr (cid:2) L ( o⋆ ) : K (cid:3) + X µ Tr (cid:2) L ( µ⋆ ) : ∂ µ K (cid:3) + X µ Tr (cid:2) L ( µ⋆ ) : A † µ K (cid:3) = 0 , X µ Tr (cid:2) L ( µ ) : Ψ δ µκ H (cid:3) + X µ Tr (cid:2) L ( µ⋆ ) : δ µκ H † Ψ † (cid:3) + − X µ Re Tr h(cid:16) P g ∂ µ ˆ G ( µκ⋆ ) A + P g { A † µ , P g ˆ G ( µκ⋆ ) A } (cid:17) † [ H ] i = 0 , ≤ κ ≤ N .
With (5.13) the 3rd set of equations can be rewritten
Re Tr h(cid:0) Ψ † ([ L ( κ ) ] † + [ L ( κ⋆ ) ] (cid:1) † H i + iRe Tr h(cid:0) i Ψ † ([ L ( κ ) ] † − [ L ( κ⋆ ) ] (cid:1) † H i + X µ Re Tr h(cid:16) P g ∂ µ ˆ G ( µκ ) A − { A µ , P g ˆ G ( µκ ) A } (cid:17) † † [ H ] i = 0 , ≤ κ ≤ N .
Because of assumption (5.14) the iRe Tr -term cancels. The assumption g † = g enables usto interchange † and P g . • Finally (5.17) follows as in the proof of Thm (5.3). (cid:4)
Finally we want to find the conservation law of ’conserved currents’.
Theorem 5.6
Consider proto-Lagrangians L and G as in Theorem 5.5. Suppose for some B ∈ g they both have the invariance properties L (P e s B ; (P e s B ) † ; . . . Q λ e s B . . . ; . . . (Q λ e s B ) † . . . ; x ) == L (P; P † ; . . . Q λ . . . ; . . . Q † λ . . . ; x ) + O ( s ) , (5.18) G ( . . . , e − s B R µν e s B , . . . ; . . . , e s B † R † θρ e − s B † , . . . ; x ) == G ( . . . , R µν , . . . ; . . . , R † θρ , . . . ; x ) + O ( s ) . (5.19) Then, the solutions to the E-L-system (5.16) satisfy the conservation law N X µ =1 ∂∂x µ n Tr h L ( µ ) ψ,A : Ψ B i + Tr h L ( µ⋆ ) ψ,A : B † Ψ † i + N X κ =1 h P g ˆ G ( µκ ) A : { A κ , B } i o = 0 . (5.20) Proof
Add the Lagrange densities L ψ,A and G A and put to the dd s of the expression L ( Ψ e s B ; e s B † Ψ † ; . . . , ∂ µ Ψ e s B + Ψ A µ e s B , . . . ; . . . , e s B † ∂ µ Ψ † + e s B † A † µ Ψ † , . . . ; x ) ++ G ( . . . , e − s B F µν e s B , . . . ; . . . , e s B † F † θρ e − s B † , . . . ; x ) One finds, Tr h L ( o ) ψ,A : Ψ B i + X µ Tr h L ( µ ) ψ,A : ∂ µ Ψ B i + X µ Tr h L ( µ ) ψ,A : Ψ A µ B i ++ Tr h L ( o⋆ ) ψ,A : B † Ψ † i + X µ Tr h L ( µ⋆ ) ψ,A : B † ∂ µ Ψ † i + X µ Tr h L ( µ⋆ ) ψ,A : B † A † µ Ψ † i ++ X µ<ν Tr h G ( µν ) A : { F µν , B } i + X θ<ρ Tr h G ( θρ⋆ ) A : { B † , F † θρ } i = 0 . (5.21)31ewrite the 3rd term and the 6th term: X µ Tr h L ( µ ) ψ,A : Ψ A µ B i = X κ Tr h L ( κ ) ψ,A : Ψ { A κ , B } i + X µ Tr h A µ L ( µ ) ψ,A : Ψ B i , X µ Tr h L ( µ⋆ ) ψ,A : ( Ψ A µ B ) † i = X κ Tr h L ( κ⋆ ) ψ,A : ( Ψ { A κ , B } ) † i + X µ Tr h A † µ L ( µ⋆ ) ψ,A : ( Ψ B ) † i . These identities, together with the 1st E-L-equation of (5.16) turn the first 6 terms of(5.21) into X µ ∂ µ Tr h L ( µ ) ψ,A : Ψ B i + X µ ∂ µ Tr h L ( µ⋆ ) ψ,A : B † Ψ † i ++ X κ Tr h L ( κ ) ψ,A : Ψ { A κ , B } i + X κ Tr h L ( κ⋆ ) ψ,A : ( Ψ { A κ , B } ) † i With Trace identity (5.13) and condition (5.14) the latter becomes X µ ∂ µ Tr h L ( µ ) ψ,A : Ψ B i + X µ ∂ µ Tr h L ( µ⋆ ) ψ,A : B † Ψ † i ++ 2 N X κ, µ =1 Re Tr h(cid:16) P g ∂ µ ˆ G ( µκ ) A − { A µ , P g ˆ G ( µκ ) A } (cid:17) : { A κ , B } i . (5.22)Next, because of (anti)symmetry, B ∈ g being constant and the definition of F µν , the final2 terms of (5.21) equal to Re N X µ,ν =1 Tr h ˆ G ( µν ) A : { F µν , B } i = Re N X µ,ν =1 Tr h ˆ G ( µν ) A : ∂ µ { A ν , B } i + − Re N X µ, ν =1 Tr h ˆ G ( µν ) A : ∂ ν { A µ , B } i − Re N X µ, ν =1 Tr h G ( µν ) A : {{ A µ , A ν } , B } i == 2Re N X µ,ν =1 Tr h ˆ G ( µν ) A : ∂ µ { A ν , B } i − Re N X µ, ν =1 Tr h G ( µν ) A : {{ A µ , A ν } , B } i . (5.23)If we add (5.22), (5.23), we arrive at (5.20), up to a term − Re N X κ, µ =1 (cid:16) h { A µ , P g ˆ G ( µκ ) A } : { A κ , B } i + Tr h P g G ( µκ ) A : {{ A µ , A κ } , B } i (cid:17) . − Re N X κ, µ =1 (cid:16) Tr h { A µ , P g ˆ G ( µκ ) A } : { A κ , B } i − Tr h { A κ , P g ˆ G ( µκ ) A } : { A µ , B } i ++ Tr h P g G ( µκ ) A : {{ A µ , A κ } , B } i (cid:17) . Each term in this sum equals because of the trace identity Tr h { M , G } : { K , B } i − Tr h { K , G } : { M , B } i + Tr h G : {{ M , K } , B } i = 0 . Indeed, note that for any M , G , K , B ∈ C c × c , Tr h MGKB − GMKB − MGBK + GMBK − KGMB + GKMB ++ KGBM − GKBM + GMKB − GKMB − GBMK + GBKM i = 0 . (cid:4) Addendum on Free Gauge Fields
If we put G A ( x ) = G ( . . . , F µν ( x ) , . . . ; . . . , F † θρ ( x ) , . . . ; x ) == G ( . . . , ∂ µ A ν − ∂ ν A µ − { A µ , A ν } , . . . ; . . . , ∂ µ A † ν − ∂ ν A † µ + { A † µ , A † ν } , . . . ; x ) == L ( A ( x ) ; A † ( x ) ; . . . , ∂ µ A ( x ) , . . . ; . . . , ∂ µ A † ( x ) , . . . ; x ) , (A.1)with A = col[ . . . , A µ , . . . ] , which now plays the role of Ψ in section 2, we get, in accordancewith our notation in section 2, L ( o ) A = row [ . . . . . . . . . . . . − P Nµ =1 { ˆ G ( µκ ) A , A µ } . . . . . . ] L (1) A = row [ 0 G ( ) A G ( ) A . . . G ( κ ) A . . . G ( N ) A ] L (2) A = row [ − G ( ) A G ( ) A . . . G ( κ ) A . . . G ( N ) A ] L (3) A = row [ − G ( ) A − G ( ) A . . . G ( κ ) A . . . G ( N ) A ] . . . = row [ . . . . . . . . . . . . . . . . . . . . . ] L ( κ ) A = row [ − G ( κ ) A − G ( κ ) A − G ( κ ) A . . . . . . G ( κN ) A ] . . . = row [ . . . . . . . . . . . . . . . . . . . . . ] L ( N ) A = row [ − G ( N ) A − G ( N ) A − G ( N ) A . . . − G ( κN ) A . . . (A.2)With convention (3.15) the lower N rows of this table simplify to L ( µ ) A = row [ . . . , ˆ G ( µκ ) A , . . . ] , ≤ µ, κ ≤ N . (A.3)Table (A.2) enables to reduce the proof of Theorem 3.2 to an application of Theorem 2.4.Because of property (3.7) it is obvious that all ’components’ of L ( µ⋆ ) A , ≤ µ ≤ N, are the hermitean transposed of the components of L ( µ ) A , ≤ µ ≤ N . Only for L ( o⋆ ) A this is notimmediately obvious. Let us check it in an ad hoc way by calculating the κ -th componentof L ( o⋆ ) A . In (A.1) replace { A † µ , A † ν } by the perturbation { A † µ + εδ µκ H , A † ν + εδ νκ H } .Now differentiate the result to ε . At ε = 0 it becomes X ≤ µ<ν ≤ N Tr h G ( µν⋆ ) A : { δ µκ H , A † ν } + { A † µ , δ νκ H } i == X κ<ν ≤ N Tr h G ( κν⋆ ) A : { H , A † ν } i + X ≤ µ<κ Tr h G ( µκ⋆ ) A : { A † µ , H } i == X κ<ν ≤ N Tr h { A † ν , G ( κν⋆ ) A } : H i + X ≤ µ<κ Tr h { G ( µκ⋆ ) A , A † µ } : H i = Tr h N X µ =1 { ˆ G ( µκ⋆ ) A , A † µ } : H i . h N X µ =1 { ˆ G ( µκ⋆ ) A , A † µ } i † = − N X µ =1 { ˆ G ( µκ ) A , A µ } . Remark on Thm 4.9-b:
If it happens that G ( . . . , e sS λµ ∂ λ A ν − e sS θν ∂ θ A µ − { A µ , A ν } , . . . ; . . . , e sS λµ ∂ λ A † ν − e sS θν ∂ θ A † µ + { A † µ , A † ν } , . . . ; x ) == G ( . . . , ∂ µ A ν − ∂ ν A µ − { A µ , A ν } , . . . ; . . . , ∂ µ A † ν − ∂ ν A † µ + { A † µ , A † ν } , . . . ; x ) + O ( s ) , it follows that Re X µ<ν Tr h G ( µν ) A : S λµ ∂ λ A ν − S θν ∂ θ A µ i = 0 . B Electromagnetism
Some more details on Example 3.4B: G A = X ≤ µ<ν ≤ ( − δ µ + δ ν Tr (cid:2) F † µν F µν (cid:3) G ( ) A = −F † G ( ) A = −F † G ( ) A = −F † G ( ) A = F † G ( ) A = F † G ( ) A = F † Now (3.19) reads, for ≤ κ ≤ , κ = 0 : ∂ G ( ) A + ∂ G ( ) A + ∂ G ( ) A == − ∂ ( ∂ A † − ∂ A † ) − ∂ ( ∂ A † − ∂ A † ) − ∂ ( ∂ A † − ∂ A † )= − ∂ ( ∂ A † + ∂ A † + ∂ A † ) + ∂ ∂ A † + ∂ ∂ A † + ∂ ∂ A † κ = 1 : − ∂ G ( ) A + ∂ G ( ) A + ∂ G ( ) A == ∂ ( ∂ A † − ∂ A † ) + ∂ ( ∂ A † − ∂ A † ) + ∂ ( ∂ A † − ∂ A † )= ∂ ∂ A † + ∂ ( − ∂ A † + ∂ A † + ∂ A † + ∂ A † ) − ( ∂ ∂ + ∂ ∂ + ∂ ∂ ) A † κ = 2 : − ∂ G ( ) A − ∂ G ( ) A + ∂ G ( ) A == ∂ ( ∂ A † − ∂ A † ) − ∂ ( ∂ A † − ∂ A † ) + ∂ ( ∂ A † − ∂ A † )= ∂ ∂ A † + ∂ ( − ∂ A † + ∂ A † + ∂ A † + ∂ A † ) − ( ∂ ∂ + ∂ ∂ + ∂ ∂ ) A † κ = 3 : − ∂ G ( ) A − ∂ G ( ) A − ∂ G ( ) A == ∂ ( ∂ A † − ∂ A † ) − ∂ ( ∂ A † − ∂ A † ) − ∂ ( ∂ A † − ∂ A † )= ∂ ∂ A † + ∂ ( − ∂ A † + ∂ A † + ∂ A † + ∂ A † ) − ( ∂ ∂ + ∂ ∂ + ∂ ∂ ) A †
35f we put A † = − Φ and col[ A † , A † , A † ] = A we get Maxwell’s equations ’in potential form’ ∂∂t div A + ∆Φ = 0 ∂ ∂t A − ∆ A + grad (cid:0) ∂∂t Φ + div A (cid:1) = 0 (B.1)If the pair A, B satisfies this pair, then the pair E = − ∂ A∂t − gradΦ , B = rot A , satisfiesthe classical homogeneous Maxwell equations: ∂ t B = rot ∂ t A = rot( − E − gradΦ) = − rot E∂ t E = ∂ t ∂ t A − grad ∂ t Φ = − ∆ A + grad div A = rot rot A = rot B Finally, imposing the ’Lorenz-Gauge’ ∂∂t
Φ + div A = 0 , we find the usual wave equationsfor Φ and A.Any solution to the system (B.1) can be reduced to a solution which satisfies the Lorentzcondition, by means of a ’gauge transform’ Φ Φ − ∂ t Λ , A A − gradΛ , leading to thesame E, B -fields. cf. Jackson [J], p.241.Similar results can be found for more general free fields governed by G = X µνθ ⋆ ρ ⋆ g µθ ⋆ g νρ ⋆ Tr h J F † θ ⋆ ρ ⋆ J − F µν i . References [DM]
W. Drechsler, M.E. Mayor: Fiber Bundle Techniques in Gauge Theories. LectureNotes in Physics 67. Springer Berlin 1977. [H]
L. Hörmander: Complex Analysis in Several Variables. North Holland Publ. Co.1973. [J]
J.D. Jackson: Classical Electrodynamics 3rd ed. John Wiley. N.J. 1998. [JP]
E.M. de Jager, H.G.J. Pijls: Proc. Sem. Mathematical Structures in Field Theories1981-1982. ISBN 90 6196 2781. CWI Amsterdam 1984. [M]
A. Messiah: Quantum Mechanics Vol II. North Holland Publ. Co. 1962. [W][W]