矩陣的行列式的幾何意義是矩陣對應的線性變換前後的面積比。
不過話說回來，講線性代數的書不一定會講到這個幾何意義，因為定義行列式幾行就寫完了，但是定義面積(體積)，尤其是高維空間的面積(體積)是一件相當麻煩的事情。
如果讀者只讀過線性代數，那麽不妨這樣直觀感受一下行列式。而如果讀者讀過實變函數或者測度論，那麽這個結論可以作為一道不錯的習題。
[PDF]Local Gauge Invariance - Nikhef
www.nikhef.nl/~h24/qcdcourse/section-3.pdf
where the generator G is to be identified later. Since ⇤ is an
arbitrary function of x and t we require that ✏ is also an arbitrary
function of x and t. Because ✏ can vary
[PDF]Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras
www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter7.pdf
[PDF]The why and how of gauge theories! - damtp
www.damtp.cam.ac.uk/user/.../Gauge_theories.p...
University of Cambridge
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Feb 24, 2015 - Those three forces are electromagnetism, the weak force and the strong force. These are described ... The gauge fields unrelated by a gauge transformation are called A/G in [1]. We can remove ... spacetime, it is called a local symmetry. ... theory. These gauge fields mediate a force between the fields ¯Ψ, Ψ.
Lawhigh - Docstoc
www.docstoc.com/docs/171120336/Lawhigh
Jun 6, 2014 - Local Gauge Transformation Covariant derivative Quantum particle equations that are locally gauge invariant Photon is massless!
[PDF]Local Gauge Invariance - Nikhef
Nov 16, 2013 - of a local gauge transformation. • Now consider the .... For massless fields we recover the Maxwell equations in empty space (no sources or ...
www.nikhef.nl/~h24/qcdcourse/section-3.pdf
Electric charge conservation
• In subatomic physics it is customary to express electric charge in
units of the elementary charge e = 1.6⇥10−19 Coulomb. The
electron then has charge −1, the positron +1, the up quark +2
3,
the down quark −1
3, etc., see the table on Page 1–5.
• As far as we know, total electric charge is the same in the initial
and final state of any elementary reaction, and this charge
conservation is experimentally verified to great accuracy.
• For instance electron decay
e ! ! ⌫e
is allowed by all known conservation laws but is forbidden by charge
conservation and it indeed has never been observed. In fact, the
life time of the electron is measured to be larger than 5⇥1026 years.
• We have seen that conserved quantities are related to symmetries
in the Hamiltonian, or the Lagrangian, so the question is now which
symmetry causes this charge conservation. Charge is obviously an
additive conserved quantity so that the symmetry transformation
must be continuous.
• In subatomic physics it is customary to express electric charge in
units of the elementary charge e = 1.6⇥10−19 Coulomb. The
electron then has charge −1, the positron +1, the up quark +2
3,
the down quark −1
3, etc., see the table on Page 1–5.
• As far as we know, total electric charge is the same in the initial
and final state of any elementary reaction, and this charge
conservation is experimentally verified to great accuracy.
• For instance electron decay
e ! ! ⌫e
is allowed by all known conservation laws but is forbidden by charge
conservation and it indeed has never been observed. In fact, the
life time of the electron is measured to be larger than 5⇥1026 years.
• We have seen that conserved quantities are related to symmetries
in the Hamiltonian, or the Lagrangian, so the question is now which
symmetry causes this charge conservation. Charge is obviously an
additive conserved quantity so that the symmetry transformation
must be continuous.
The answer, as we will see, is that a so-called gauge symmetry
is responsible for the charge conservation. Gauge transformations
enter when interactions are described in terms of potentials, instead
of forces. A well known example is from classical electrodynamics
where we can transform the scalar and vector potentials in such a
way that the E and B fields are una↵ected.
is responsible for the charge conservation. Gauge transformations
enter when interactions are described in terms of potentials, instead
of forces. A well known example is from classical electrodynamics
where we can transform the scalar and vector potentials in such a
way that the E and B fields are una↵ected.
[PDF]8. Gauge theories
Thus local U(1) gauge invariance requires the existence of a massless ... tions of n2 − 1 gauge bosons with matter, using as a single parameter the gauge coupling g. ... Since the number of generators is m = n2 − 1 for SU(n), the groups SU(2).
http://physics.stackexchange.com/questions/65141/do-generators-belong-to-the-lie-group-or-the-lie-algebra
web.phys.ntnu.no/~mika/skript_qft7.pdf
http://physics.stackexchange.com/questions/65141/do-generators-belong-to-the-lie-group-or-the-lie-algebra
[PDF]Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras
www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter7.pdf
Consider a set of elements R that depend on a number of real continuous
parameters, R(a) ´ R(a1; a2; : : : ; ar). These elements are said to form
a continuous group if they ful¯ll the requirements of a group (Section
2.1) and if there is some notion of `proximity' or `continuity' imposed
on the elements of the group in the sense that a small change in one
of the factors of a product produces a correspondingly small change in
their product. If the group elements depend on r parameters, this is
called an r-parameter continuous group.
In general terms, the requirements that a continuous set of elements
form a group are the same as those for discrete elements, namely, closure
under multiplication, associativity, the existence of a unit, and
an inverse for every element.
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