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Jun
17
awarded  Yearling
Jun
17
comment Clarification on meaning of scalar in math and scalar in physics
But doesn't this fact (about general coordinate transformations, or change of basis, or whatever) follow from the mathematician's definition? So how could they be equivalent?...
Jun
17
comment Clarification on meaning of scalar in math and scalar in physics
But wouldn't a physicist have to check all possible coordinate transformations in order to know if the quantity was a scalar? Why just rotations and reflections? I know they are isometries, but what if I do some weird transformation that gives me something I wouldn't expect?
Jun
17
comment Clarification on meaning of scalar in math and scalar in physics
@DavidZ Ok that's my question, are the definitions equivalent?
Jun
17
comment Clarification on meaning of scalar in math and scalar in physics
@David But is this not weird? Group actions have nothing to do with scalars in the mathematicians' sense.
Jun
17
comment Clarification on meaning of scalar in math and scalar in physics
So suppose an object satisfies 1. but not 2., then it can't be interpreted as a thing which doesn't vary with coordinate changes because otherwise it would have to satisfy 2...but what about every other possible coordinate transformation I can talk about that you haven't mentioned in 1. and 2.? What if object x is a scalar under 1. and 2., hence a scalar by the mathematicians definition, but then I introduce condition 3. which x doesn't satisfy? Then it wouldn't be a scalar under 1. 2. and 3., but then how could it be a scalar with the math definition?
Jun
17
comment Clarification on meaning of scalar in math and scalar in physics
@aaaaaa I think it's just a terminology issue.
Jun
17
asked Clarification on meaning of scalar in math and scalar in physics
Jun
15
awarded  Popular Question
Feb
20
comment Variational Principle to find Energy Eigenfunctions
$a=H\psi\,,b=\psi\,.$ Note that $\langle H\psi,H\psi\rangle=\|H\psi\|^2\,.$ So we have $|\langle H\psi,\psi\rangle|\le\|H\psi\|\|\psi\|=\|H\psi\|\,.$
Feb
20
comment Variational Principle to find Energy Eigenfunctions
The Cauchy-Schwarz inequality says that $|\langle a,b\rangle|\le\|a\|\|b\|$ with equality iff $a=\lambda b$ where $\lambda$ is a scalar.
Feb
19
revised Variational Principle to find Energy Eigenfunctions
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Feb
19
revised Variational Principle to find Energy Eigenfunctions
added 178 characters in body
Feb
19
revised Variational Principle to find Energy Eigenfunctions
added 178 characters in body
Feb
19
answered Variational Principle to find Energy Eigenfunctions
Feb
2
comment Every Relativistic Field Satifies the Klein-Gordon Equation?
Ok thanks. One more question: Is there a sense in which quantum fields satisfy the Klein-Gordon equation? I mean since they are operators. I've read that particles satisfy the Klein-Gordon equation, something about their mass being on-shell, but it's not really clear to me how to go from a field to a particle, some confusion resulting by the presence of real particles and virtual particles. Though I know that light (or electromagnetic waves) satisfies the Klein-Gordon equation with $m=0\,.$
Feb
1
comment Every Relativistic Field Satifies the Klein-Gordon Equation?
Without forces? The analogous classical mechanics Lagrangian is $\frac{\dot{x}^2}{2}-kx^2\,,$ equivalently $F=kx\,,$ so doesn't the term involving $m$ result in a force of some kind? Is the definition of noninteracting "linear"?
Feb
1
comment Every Relativistic Field Satifies the Klein-Gordon Equation?
What's the definition of a free field? Is it defined to be a field with that Lagrangian density? This is how I've seen it defined. Which would make it a tautology...
Feb
1
asked Every Relativistic Field Satifies the Klein-Gordon Equation?
Jan
3
comment Is conservation of energy a set of principles that is inevitable in any 'possible world'?
You can look up the variational complex, or look at Peter Olver's book "Applicatons of Lie Groups to Differential Equations". It's similar to how you determine if a function is the gradient of a scalar actually. I've written a bit about it, but it's not the best exposition.