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Dynamical symmetry transformations

Classical mechanics

Consider the transformation where the change of coordinates also depends on the generalized velocities.$${\displaystyle {\begin{aligned}q^{r}\to q^{r}+\delta q^{r}\\\delta q^{r}=\epsilon \phi ^{r}(q,{\dot {q}},t)\\\end{aligned}}}$$ If the above is a dynamical symmetry, then the lagrangian changes by:$${\displaystyle \delta L=\epsilon {\frac {d}{dt}}F(q,{\dot {q}},t)}$$ and the new Lagrangian is said to be dynamically equivalent to the old Lagrangian as it ensures the resultant equations of motion being the same.

Using the change in Lagrangian property of a dynamical symmetry:$${\displaystyle {\frac {d}{dt}}F={\frac {\partial F}{\partial q^{r}}}{\dot {q}}^{r}+{\frac {\partial F}{\partial {\dot {q}}^{r}}}{\ddot {q}}^{r}+{\frac {\partial F}{\partial t}}={\frac {\delta L}{\epsilon }}=\left({\frac {\partial L}{\partial q^{r}}}\phi ^{r}+{\frac {\partial L}{\partial {\dot {q}}^{r}}}{\frac {\partial \phi ^{r}}{\partial t}}\right)+p_{s}{\frac {\partial \phi ^{s}}{\partial q^{r}}}{\dot {q}}^{r}+p_{s}{\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{r}}}{\ddot {q}}^{r}}$$ Since the ${\displaystyle {\ddot {q}}}$  terms appear only once in either side, it's coefficients must be equal for this to be true, giving the relation:$${\displaystyle p_{s}{\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{r}}}={\frac {\partial F}{\partial {\dot {q}}^{r}}}}$$ The above relation is useful in dervations involving infinitesimal generators of transformations.

Classical Field theory

Consider the symmetry transformations of fields in classical field theory as:

${\textstyle {\begin{aligned}x\rightarrow &\,x'=x+\delta x(x)\\\phi (x)\rightarrow &\,\phi '(x')=\phi (x)+\delta \phi (x)\end{aligned}}}$ 

Since the parameters can change, that is ${\textstyle x\rightarrow x'=x+\delta x(x)}$ , it is no longer sufficient to consider only change in Lagrangians independently as the volume element also contributes changes, nonetheless the same can be shown using change in action. The change in volume element is given by

${\displaystyle \det({\frac {\partial x'}{\partial x}})=|(1+\partial _{0}\delta x^{0})(1+\partial _{1}\delta x^{1})\cdots (1+\partial _{n-1}\delta x^{n-1})|+{\mathcal {O}}(\delta x^{2})=1+\partial _{\mu }\delta x^{\mu }+{\mathcal {O}}(\delta x^{2})}$ 

The terms arising in change in action is collected into effective change in Lagrangian as ${\displaystyle \delta \int {\mathcal {L}}\,d^{n}x=\int {\mathcal {L}}\,\delta (d^{n}x)+\int \delta {\mathcal {L}}\,d^{n}x=:\int \delta {\mathcal {L}}_{\text{eff}}\,d^{n}x}$  which gives ${\textstyle \delta {\mathcal {L}}_{\text{eff}}={\mathcal {L}}\,\partial _{\mu }\delta x^{\mu }+\delta {\mathcal {L}}}$ . Hence the modified conditions for dynamical symmetry are given as ${\textstyle \delta L_{\text{eff}}=L\,\partial _{\mu }\delta x^{\mu }+\delta L=\partial _{\mu }F^{\mu }}$ .

Generalized Noether theorem on Dynamical symmetries

Noether Invariant in classical mechanics

If Euler Lagrange relation is satisfied for the provided Lagrangian, the invariants of motion can be derived as:$${\displaystyle \delta L-\epsilon {\frac {d}{dt}}F(q,{\dot {q}},t)=\epsilon \phi {\cancelto {=0}{\left({\frac {\partial }{\partial q}}-{\frac {d}{dt}}{\frac {\partial }{\partial {\dot {q}}}}\right)L}}+\epsilon {\frac {d}{dt}}\left(\phi {\frac {\partial }{\partial {\dot {q}}}}L-F\right)=\epsilon {\frac {d}{dt}}\left(\phi {\frac {\partial }{\partial {\dot {q}}}}L-F\right)=0}$$ Hence ${\displaystyle \left(\phi {\frac {\partial }{\partial {\dot {q}}}}L-F\right)=p\phi -F}$  is a constant of motion. Since Euler Lagrange equation is satisfied, the derived Noether invariant also generates the same symmetry transformation.

Update this with noether theorem from "Lagrangian Interaction An Introduction to Relativistic Symmetry in Electrodynamics and Gravitation - Doughty" including change in time.

Conserved current in classical field theory

Consider the symmetry transformations of fields in classical field theory as:

${\textstyle {\begin{aligned}x\rightarrow &\,x'=x+\delta x(x)\\\phi (x)\rightarrow &\,\phi '(x')=\phi (x)+\delta \phi (x)\end{aligned}}}$ 

The total variation of fields is defined as ${\textstyle \Delta \phi \equiv \phi '(x)-\phi (x)=-\partial _{\mu }\phi (x)\delta x^{\mu }+\delta \phi (x)}$  upto first order. Similarly it also follows that ${\textstyle \Delta \partial \phi =\partial \Delta \phi }$ .

${\displaystyle \delta L={\frac {\partial L}{\partial \phi }}\Delta \phi (x)+{\frac {\partial L}{\partial \partial \phi }}\Delta \partial \phi +\partial _{\mu }L\delta x^{\mu }=\left({\frac {\partial L}{\partial \phi }}-\partial {\frac {\partial L}{\partial \partial \phi }}\right)\Delta \phi (x)+\partial _{\mu }(\pi ^{\mu }\Delta \phi +L\delta x^{\mu })-L\partial _{\mu }\delta x^{\mu }}$ 

Using equations of motion i.e. in the on-shell condition and combining terms in the symmetry condition ${\textstyle \delta L_{\text{eff}}=L\,\partial _{\mu }\delta x^{\mu }+\delta L=\partial _{\mu }F^{\mu }}$ , the conservation of current is derived as:

${\textstyle \partial _{\mu }(\pi ^{\mu }\Delta \phi +L\delta x^{\mu }-F^{\mu })=\partial _{\mu }j^{\mu }=0}$ 

Using ${\textstyle \partial _{\mu }L={\frac {\partial L}{\partial \phi }}\partial _{\mu }\phi +{\frac {\partial L}{\partial \partial _{\nu }\phi }}\partial _{\mu }\partial _{\nu }\phi +\partial _{\mu }^{\text{ex}}L}$  the effective change can be expressed as:

${\textstyle \delta L_{\text{eff}}={\frac {\partial L}{\partial \phi }}\Delta \phi (x)+{\frac {\partial L}{\partial \partial \phi }}\Delta \partial \phi +\partial _{\mu }(L\delta x^{\mu })={\frac {\partial L}{\partial \phi }}\delta \phi +\pi ^{\nu }(-\partial _{\mu }\phi \partial _{\nu }\delta x^{\mu }+\partial _{\nu }\delta \phi )+\partial _{\mu }^{\text{ex}}L\delta x^{\mu }+L\partial _{\mu }\delta x^{\mu }}$ 

which can be used to check for symmetry of transformation i.e. if it can be expressed as a divergence of some vector function ${\textstyle \delta L_{\text{eff}}=\partial _{\mu }F^{\mu }}$  under certain broad assumptions on the form of Lagrangian.^[1]

Conserved charge in classical field theory

If the spatial components of conserved current vanish at the boundary, ${\textstyle \int \partial _{\mu }j^{\mu }\,d^{3}x=0=\int \partial _{0}j^{0}\,d^{3}x=\partial _{0}(\int j^{0}\,d^{3}x)}$ is corresponding conserved charge.

Summary of generalized Noether theorem for Dynamical symmetries

Noether's first theorem

The symmetry condition for both classical Lagrangian and field theory Lagrangians can be expressed without making mention of the function, ${\textstyle F}$  as follows:^[2]

1. For classical field theories: ${\textstyle \left({\frac {\partial L}{\partial \phi }}-\partial _{\mu }{\frac {\partial L}{\partial \partial _{\mu }\phi }}\right)\Delta \phi \,\equiv \,\partial _{\mu }j^{\mu }}$ 
2. For classical Lagrangian mechanics: ${\textstyle \left({\frac {\partial L}{\partial q}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}}}\right)\delta q\,\equiv \,{\frac {d}{dt}}Q}$ 

that it is possible to simplify the following off-shell quantities as divergence of some function.

The above equations are known as strong equations (also refered to as kinematic or improper) which unlike weak equations (also refered to as dynamical or proper), are relations that are simplified without invoking the equations of motion or Euler Lagrange equations. According to convention related to these relations, equivalence symbol is used for strong relations and "is equal to" symbol for weak relations. Only the symmetry condition for variations ${\textstyle \delta q}$  or ${\textstyle \Delta \phi }$ , which is expressed as ${\textstyle \delta L_{\text{eff}}=\partial _{\mu }F^{\mu }}$  or the invariance of action, is used for the strong relations.

Since Euler Lagrange equations are satisfied on-shell, the corresponding ${\textstyle {\frac {d}{dt}}Q}$  and ${\textstyle \partial _{\mu }j^{\mu }}$  also vanish on-shell. The weak form of the above relations is hence derived by combining with on-shell EOM relations, resulting in Noether's first theorem conserved current relations: ${\textstyle \partial _{\mu }j^{\mu }=0,\quad {\frac {dQ}{dt}}=0}$ .

Noether's second theorem

The same relations occur with additional terms in the case of local symmetries, which is a class of symmetries with infinite generators parametrized in a certain manner by arbitrary functions instead of arbitrary constants, leading to Noether's second theorem. In noether's second theorem, local symmetries are involved where ${\textstyle \delta x=0,\Delta \phi =\Delta \phi [f,\partial _{\mu }f,\partial _{\mu }\partial _{\nu }f,\cdots ;x]}$  which is a symmetry of the system for any arbitrary function ${\textstyle f}$ . Hence, the derived strong relations are of the following form:

${\textstyle \left({\frac {\partial L}{\partial \phi }}-\partial _{\mu }{\frac {\partial L}{\partial \partial _{\mu }\phi }}\right){\frac {\delta \Delta \phi }{\delta f}}\,\equiv \,\partial _{\mu }\left(\left({\frac {\partial L}{\partial \phi }}-\partial _{\nu }{\frac {\partial L}{\partial \partial _{\nu }\phi }}\right){\frac {\delta \Delta \phi }{\delta \partial _{\mu }f}}\right)-\partial _{\mu }\partial _{\nu }\left(\left({\frac {\partial L}{\partial \phi }}-\partial {\frac {\partial L}{\partial \partial \phi }}\right){\frac {\delta \Delta \phi }{\delta \partial _{\mu }\partial _{\nu }f}}\right)+\cdots }$ 

Klien-Utiyama theorem/Noether's third theorem

The strong form of Noether's first theorem can be combined with that of the second theorem to get another strong relation, particularly since ${\textstyle f(x)=k}$  should have its associated Noether's first theorem strong relations with ${\textstyle \Delta \phi =k{\frac {\delta \Delta \phi }{\delta f}}}$  where the constant can be absorbed into the conserved current.^[3][2]

${\textstyle \partial _{\mu }\left(j^{\mu }-\left({\frac {\partial L}{\partial \phi }}-\partial _{\nu }{\frac {\partial L}{\partial \partial _{\nu }\phi }}\right){\frac {\delta \Delta \phi }{\delta \partial _{\mu }f}}+\partial _{\nu }\left(\left({\frac {\partial L}{\partial \phi }}-\partial {\frac {\partial L}{\partial \partial \phi }}\right){\frac {\delta \Delta \phi }{\delta \partial _{\mu }\partial _{\nu }f}}\right)+\cdots \right)\,\equiv \,0}$ 

which is a strong relation, showing that the divergence should vanish identically for the function. Gauge symmetries for which these results are expected provide a stronger set of conditions to be satisfied for the symmetry. For example, the gauge symmetry of fields in QED Lagrangian of spin half field demands continuity equation for ${\textstyle j^{\mu }}$  which is a stronger conditition than that derived in Noether's first theorem, since it does not assume the form of equations of motion or Euler Lagrange equations.^[2] Hence gauge symmetry conditions can be used to find strong relations relating some conserved current.

Add also quasi-invariance of Lagrangian, consequences with singular Lagrangians, etc from Rothe n Rothe.

Generator of transformation

Using the calculated relations, the following Poisson brackets can be computed as:$${\displaystyle \{q^{r},\epsilon (p_{s}\phi ^{s}-F)\}=\delta q^{r},\quad \{p_{r},\epsilon (p_{s}\phi ^{s}-F)\}=\delta p_{r}+\epsilon \left({\frac {\partial L}{\partial q^{s}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}^{s}}}\right){\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{r}}}}$$ Hence, the term ${\displaystyle p\phi -F}$  generates the canonical dynamical symmetry transformation if either the Euler Lagrange relation is satisfied, or if ${\displaystyle {\frac {\partial \phi _{s}}{\partial {\dot {q}}^{r}}}=0\,\forall s,r}$  which is a infinitesimal point transformation. Note that in the point transformation condition, the quantity generates the transformation regardless of if the Euler Lagrange equations are satisfied and since they do not depend on the dynamics of the problem are said to be a purely kinematic relation.^[4]

Similar results are obtained in classical field theory, for example, in a Lorentz invariant Lagrangian density where corresponding conserved charges, momentum density ${\displaystyle P^{\mu }}$  generates translation of fields and ${\displaystyle M^{\mu \nu }}$ of Lorentz invariance generates Lorentz transformation of fields.^[5]

Proof

The change in generalized velocity and momentum term can be derived as: $${\displaystyle {\begin{aligned}p={\frac {\partial L}{\partial {\dot {q}}}},\quad &{\dot {q}}={\frac {dq}{dt}}\\\delta p_{r}={\frac {\partial ^{2}L}{\partial q^{s}\partial {\dot {q}}^{r}}}\delta q^{s}+{\frac {\partial ^{2}L}{\partial {\dot {q}}^{s}\partial {\dot {q}}^{r}}}\delta {\dot {q}}^{s},\quad &\delta {\dot {q}}^{r}=\epsilon {\frac {\partial \phi ^{r}}{\partial q^{s}}}{\dot {q}}^{s}+\epsilon {\frac {\partial \phi ^{r}}{\partial {\dot {q}}^{s}}}{\ddot {q}}^{s}+\epsilon {\frac {\partial \phi ^{r}}{\partial t}}\\\end{aligned}}}$$  Firstly, the change in momentum can be expressed in a more useful form as follows:$${\displaystyle \delta p_{r}={\frac {\partial ^{2}L}{\partial q^{s}\partial {\dot {q}}^{r}}}\delta q^{s}+{\frac {\partial ^{2}L}{\partial {\dot {q}}^{s}\partial {\dot {q}}^{r}}}\delta {\dot {q}}^{s}={\frac {\partial }{\partial {\dot {q}}^{r}}}\left({\frac {\partial L}{\partial q^{s}}}\delta q^{s}+{\frac {\partial L}{\partial {\dot {q}}^{s}}}\delta {\dot {q}}^{s}\right)-{\frac {\partial L}{\partial q^{s}}}{\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta q^{s})-{\frac {\partial L}{\partial {\dot {q}}^{s}}}{\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta {\dot {q}}^{s})={\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta L)-p_{s}{\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta {\dot {q}}^{s})-{\frac {\partial L}{\partial q^{s}}}{\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta q^{s})}$$  Simplifying the required poisson brackets, ${\displaystyle {\begin{aligned}\{q^{r},\epsilon (p_{s}\phi ^{s}-F)\}=\epsilon \left(\phi _{r}+{\frac {\partial {\dot {q}}^{m}}{\partial p_{r}}}{\cancelto {=0}{\left(p_{s}{\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{m}}}-{\frac {\partial F}{\partial {\dot {q}}^{m}}}\right)}}\right)&=\delta q^{r}\\\{p_{r},\epsilon (p_{s}\phi ^{s}-F)\}=\epsilon \left(-p_{s}{\frac {\partial \phi ^{s}}{\partial q^{r}}}+{\frac {\partial F}{\partial q^{r}}}+{\cancelto {=0}{\left({\frac {\partial F}{\partial {\dot {q}}^{m}}}-p_{s}{\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{m}}}\right)}}\left({\frac {\partial {\dot {q}}^{m}}{\partial q^{r}}}\right)_{q,p,t}\right)&=\epsilon \left(-p_{s}{\frac {\partial \phi ^{s}}{\partial q^{r}}}+{\frac {\partial F}{\partial q^{r}}}\right)\\\end{aligned}}}$  As a preliminary result, for any function of ${\displaystyle (q,{\dot {q}},t)}$ , ${\displaystyle {\frac {\partial }{\partial {\dot {q}}^{r}}}{\frac {d}{dt}}-{\frac {d}{dt}}{\frac {\partial }{\partial {\dot {q}}^{r}}}={\frac {\partial }{\partial q^{r}}}+{\frac {\partial {\ddot {q}}^{s}}{\partial {\dot {q}}^{r}}}{\frac {\partial }{\partial {\dot {q}}^{s}}}}$  which can be used to calculated the quantity: ${\displaystyle {\frac {\partial }{\partial {\dot {q}}^{r}}}\left({\frac {dF}{dt}}\right)-p_{s}\left({\frac {\partial }{\partial {\dot {q}}^{r}}}\left({\frac {d}{dt}}\phi ^{s}\right)\right)-{\dot {p}}_{s}{\frac {\partial }{\partial {\dot {q}}^{r}}}(\phi ^{s})={\frac {d}{dt}}{\cancel {\left({\frac {\partial }{\partial {\dot {q}}^{r}}}F-p_{s}{\frac {\partial }{\partial {\dot {q}}^{r}}}\phi ^{s}\right)}}+{\frac {\partial {\ddot {q}}^{s}}{\partial {\dot {q}}^{r}}}{\cancel {\left({\frac {\partial }{\partial {\dot {q}}^{s}}}F-p_{m}{\frac {\partial }{\partial {\dot {q}}^{s}}}\phi ^{m}\right)}}-p_{s}{\frac {\partial \phi ^{s}}{\partial q^{r}}}+{\frac {\partial F}{\partial q^{r}}}=\{p_{r},(p\phi -F)\}}$  This relation can be restated and combined with the formula for ${\displaystyle \delta p_{r}}$ to give the required relation for momentum. ${\displaystyle \{p_{r},\epsilon (p_{s}\phi ^{s}-F)\}={\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta L)-p_{s}{\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta {\dot {q}}^{s})-{\dot {p}}_{s}{\frac {\partial }{\partial {\dot {q}}^{r}}}(\delta q^{s})=\delta p_{r}+\epsilon \left({\frac {\partial L}{\partial q^{s}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}^{s}}}\right){\frac {\partial \phi ^{s}}{\partial {\dot {q}}^{r}}}}$

A note on formalisms: De-Donder Weyl to Traditional formalism

In field theory there are many reasons to avoid De-Donder weyl formalism where dynamical variables are ${\textstyle \phi ,\pi _{\mu }={\frac {\partial L}{\partial \partial _{\mu }\phi }},x^{\mu }}$  since the number of parameters have changed, the Hamiltonian equation of motions can no longer be of the same form. Due to non-standard form of the EOM in Hamiltonian formalism, Hamiltonian also cannot be said to be a generator of a time translation and hence does not correspond to energy. It is also convinient to do calculations with known relations and reduced number of momentum terms ${\textstyle \phi ,\pi ={\frac {\partial L}{\partial {\dot {\phi }}}},t}$  refered to as Traditional formalism.

Note that this trend is reversed in the Euler Lagrange formalism, where although both formalism are feasible although, the non-removability of ${\textstyle \nabla \phi }$  term from the Lagrangian owing to its Lorentz invariance property, can make functional derivative unavoidable in the ${\textstyle \phi ,{\dot {\phi }},t}$  formalism, for simplicity it is a gradient functional on the field ${\textstyle \nabla \phi [\phi ]}$ , as opposed to ${\textstyle \phi ,{\dot {\phi }},\nabla \phi ,x,t}$  De-Donder Weyl approach.

One can make the switch from the covariant De-Donder Weyl formalism to a more traditional formalism where only time evolution is considered. This also has the benifit of providing the Hamiltonian as a generator of time translation. This is done using ${\textstyle {\mathcal {L}}[\phi ,\partial \phi ,x]\rightarrow L[\phi ,{\dot {\phi }},t]\equiv \int \,{\mathcal {L}}[\phi ,{\dot {\phi }},\partial _{i}\phi [\phi ],\mathbf {x} ,t]\,d^{3}\mathbf {x} }$ . Since both formalisms relate to minimization of the same action, they are said to be equivalent. The equations of motion are changed as follows:

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \phi }}=\partial _{\mu }{\frac {\partial L}{\partial \partial _{\mu }\phi }}\rightarrow {\frac {\partial L}{\partial \phi }}={\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}}$ 

In this formalism, the boundary conditions ${\textstyle \phi _{i}(x)=\phi (x,t_{i})\rightarrow \phi _{f}(x)=\phi (x,t_{f})}$  are given and the solution of the equations of motion relating ${\textstyle \phi (x,t)}$  are such that they satisfy the given boundary condition on ${\textstyle t=t_{i},t=t_{f}}$  and for all ${\textstyle x}$ . The relating Poisson brackets are also hence, defined in terms of the coordinates, ${\textstyle \phi }$  and respective conjugate momentum ${\textstyle \pi ={\frac {\partial L}{\partial {\dot {\phi }}}}}$ .

Corresponding ${\textstyle \partial _{\mu }F^{\mu }[\phi ,\partial \phi ,x]\rightarrow \int \,\partial _{\mu }F^{\mu }[\phi ,{\dot {\phi }},\partial _{i}\phi [\phi ],\mathbf {x} ,t]\,d^{3}\mathbf {x} =\partial _{0}\int F^{0}\,d^{3}\mathbf {x} \equiv \partial _{0}F}$  given that the vector component approaches zero on the boundaries. If ${\textstyle \delta {\mathcal {L}}_{\text{eff}}=0\rightarrow \partial _{0}F=0}$ , hence in such conditions ${\textstyle F}$  corresponds to conserved charge in the De-Donder Weyl formalism.

${\textstyle \delta L_{\text{eff}}=\int ({\mathcal {L}}\partial _{\mu }\delta x^{\mu }+\delta {\mathcal {L}})\,d^{3}x=\int {\mathcal {L}}\partial _{0}\delta x^{0}\,d^{3}x+\int \delta ({\mathcal {L}}\,d^{3}x)=\int {\mathcal {L}}\partial _{0}\delta x^{0}\,d^{3}x+\delta L=\partial _{0}F}$ 

While conserved charge relations are:

${\textstyle G=\int j^{0}\,d^{3}x=\int \pi ^{0}\delta \phi \,d^{3}x-\int \pi ^{0}\partial _{\mu }\delta x^{\mu }\,d^{3}x+\int {\mathcal {L}}\delta x^{0}\,d^{3}x-F}$ 

${\textstyle \partial _{0}G=\int \partial _{0}\pi ^{0}\delta \phi \,d^{3}x+\int \pi ^{0}\partial _{0}\delta \phi \,d^{3}x-\int \partial _{0}\pi ^{0}\partial _{\mu }\delta x^{\mu }\,d^{3}x-\int \pi ^{0}\partial _{0}\partial _{\mu }\delta x^{\mu }\,d^{3}x+\int \partial _{0}{\mathcal {L}}\delta x^{0}\,d^{3}x-\delta L}$ 

References

1. Williams, Anthony G. (2022). Quantum field theory: classical mechanics to gauge field theories. Cambridge, United Kingdom ; New York, NY: Cambridge University Press. ISBN 978-1-108-47090-2.
2. Brading, Katherine; Brown, Harvey R. (2000-09-08), Noether's Theorems and Gauge Symmetries, arXiv, doi:10.48550/arXiv.hep-th/0009058, arXiv:hep-th/0009058, retrieved 2025-06-26
3. Brading, Katherine; Brown, Harvey R. (2003-12-04), "Symmetries and Noether's theorems", Symmetries in Physics, Cambridge University Press, pp. 89–109, ISBN 978-0-521-82137-7, retrieved 2025-06-17
4. Mallesh, K. S.; Chaturvedi, Subhash; Balakrishnan, V.; Simon, R.; Mukunda, N. (2011-02-01). "Symmetries and conservation laws in classical and quantum mechanics". Resonance. 16 (2): 129–151. doi:10.1007/s12045-011-0020-5. ISSN 0973-712X.
5. Greiner, Walter; Reinhardt, Joachim (1996). Field Quantization. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 49–54. doi:10.1007/978-3-642-61485-9. ISBN 978-3-540-78048-9.