Breadcrumbs Section. Click here to navigate to respective pages.

Chapter

Chapter

# Noether's Theorem and Gauge Theories of the First and Second Kinds

DOI link for Noether's Theorem and Gauge Theories of the First and Second Kinds

Noether's Theorem and Gauge Theories of the First and Second Kinds book

# Noether's Theorem and Gauge Theories of the First and Second Kinds

DOI link for Noether's Theorem and Gauge Theories of the First and Second Kinds

Noether's Theorem and Gauge Theories of the First and Second Kinds book

Click here to navigate to parent product.

## ABSTRACT

Perhaps the main point of Lagrangian formalism is that it provides a natural framework for the quantum mechanical implementation of symmetries. This is caused by the principle of stationary action taking the form of a variational principle in the dynamical equations of the Lagrangian formalism. Consider any infinitesimal transformation of the fields
Ψ
k
(
x
¯
)
→
Ψ
k
(
x
¯
)
+
i
ε
F
k
(
x
¯
)
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/2ceefd23-c448-493c-8d05-0bac4f541560/content/math7_1_B.tif"/>
which leaves the action
I
[
Ψ
]
≡
∫
∞
∞
d
t
L
[
Ψ
(
t
)
,
Ψ
˙
(
t
)
]
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/2ceefd23-c448-493c-8d05-0bac4f541560/content/math7_2_B.tif"/>
invariant. Under an arbitrary variation of Ψ(x) we get
δ
I
[
Ψ
]
=
∫
∞
∞
d
t
∫
d
3
x
¯
[
δ
L
δ
Ψ
k
δ
Ψ
k
(
x
¯
)
+
δ
L
δ
Ψ
˙
k
(
x
¯
)
]
.
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/2ceefd23-c448-493c-8d05-0bac4f541560/content/math7_3_B.tif"/>
Now assume that δΨ^{
k
} (
x
) vanishes for t → ∞ so that we may integrate by parts, and write
δ
I
[
Ψ
]
=
∫
d
r
x
¯
[
δ
L
δ
Ψ
k
(
x
¯
)
−
d
d
t
δ
L
δ
Ψ
˙
(
x
¯
)
]
δ
I
Ψ
k
(
x
¯
)
.
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/2ceefd23-c448-493c-8d05-0bac4f541560/content/math7_4_B.tif"/>
We see that the action is stationary with respect to all variations δΨ^{
k
} that vanish at t → ∞ if and only if the field satisfies the field equations
Π
˙
k
(
x
¯
,
t
)
=
∂
L
[
Ψ
(
t
)
,
Ψ
˙
(
t
)
]
δ
Ψ
k
(
x
¯
,
t
)
.
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/2ceefd23-c448-493c-8d05-0bac4f541560/content/math7_5_B.tif"/>