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3 years ago The eigenvalues must be positive for boundedness potential from below, then according to this eigenvalues I can get some constraints on parameters.
No, not all of them. Only if they have a real, non-trivial solution. Let me state the algorithm again.
- Derive the fourth rank tensor Q.
- Calculate eigenvalues via the resultant method.
- Potential is bounded iff for all real eigenvalues, all non-trivial, real solutions (i.e. eigenvectors) belong to a positive eigenvalue.
Or equivalently: The potential is not bounded from below iff a real and non-positive (i.e. zero or negative) eigenvalue has a real solution different than x=y=...=0.
Let's discuss some implications of this:
- Complex eigenvalues (yes, they do exist for symmetric tensors with rank > 2) are completely irrelevant, i.e. don't influence any parameter constraint.
- A negative eigenvalue is only relevant iff there exists a real solution of the eigenvalue equation which is not trivial (= not zero). For instance, you can have negative eigenvalues but the tensor is still positive definite if these negative eigenvalues only have complex solutions (= complex eigenvectors).
I have 4 eigenvalues according to BFB package; [...] and we know that eigenvalues must be positive
No, see above. Some eigenvalues might be irrelevant for the positivity condition.
In Positivity Test, I myself must give some numbers to parameters? For example lh = 0.129, ls = 3, lsh = 2, and run it?
Yes, it only works if all parameters are numbers. This is because PositivityTest is the most general algorithm that is guaranteed to give you an answer about the positivity of your tensor/potential (if you have enough time to wait for its conclusion). In general a positivity condition cannot be written down as inequalities (i.e. algebraic expressions). The only way to check a parameter set is to do it numerically (that's why all parameters have to be numbers). This is a direct result of the Abel-Ruffini-Theorem.
Some potentials like two Higgs doublet models (2HDMs) are an exception to this rule. They do have an algebraic expression for positivity (i.e. inequalities for their parameters).
Let me state this again: Inequalities for positivity constraints are not guaranteed to exist. They are rather an exception than a regularity (do note that we are always talking about necessary and sufficient conditions. Sufficient conditions are much more relaxed and can have inequality expressions because they are not exact, i.e. don't give the full parameter ranges for positivity).
And for last question, where is the output of Positivity Test?
Take
potential = a*x^4 + b*x^2*y^2 + c*y^4then
PositivityTest[potential /. {a -> 2, b -> 5, c -> 2}, {x, y}]gives you
<|"ErrorCode" -> 0, "PositiveDefinite" -> True, "PositiveSemidefinite" -> True|>which tells you that the derived tensor Q is positive definite -> this potential is bounded from below.
I think it is very obvious and Positivity Test is not necessary, am I wrong?
Yes and no. This depends on your definition of triviality. Let's have a look at the symbolic eigenvalues:
eigvals = {a, c, 1/2 (a + c - Sqrt[a^2 + b^2 - 2 a c + c^2]), 1/2 (a + c + Sqrt[a^2 + b^2 - 2 a c + c^2])}For a = 2, b = 5, c = 2 we have
{2, 2, -(1/2), 9/2}Notice that the third eigenvalue is negative! This is OK, because there is no real solution other than x=y=0. Hence this eigenvalue is irrelevant for judging positivity. Not so trivial after all ;)
EDIT: The example with a = 1.1, b = -3, c = 2 gave a wrong result due to numerical constraints. There is some numerical uncertainty in PositivityTest because Mathematica's solvers are ... not perfect ...
Compare
Solve[
{a x^3 - 1/2 (a + c - Sqrt[a^2 + b^2 - 2 a c + c^2]) x^3 + 1/2 b x y^2 == 0,
1/2 b x^2 y + c y^3 - 1/2 (a + c - Sqrt[a^2 + b^2 - 2 a c + c^2]) y^3 == 0}
/. {a -> 1.1, b -> -3, c -> 2}, {x, y}
]to
Solve[
{a x^3 - 1/2 (a + c - Sqrt[a^2 + b^2 - 2 a c + c^2]) x^3 + 1/2 b x y^2 == 0,
1/2 b x^2 y + c y^3 - 1/2 (a + c - Sqrt[a^2 + b^2 - 2 a c + c^2]) y^3 == 0}
/. {a -> 11/10, b -> -3, c -> 2}, {x, y}
]
Layla-Kalhor
Dear Marcel,
constraints on parameters I looked at your article and thesis again, I am a little confused about positivity test and constraints for(on) parameters. Let me see if I've got it right... The eigenvalues must be positive for boundedness potential from below, then according to this eigenvalues I can get some constraints on parameters.
For example consider this simple potential;
I have 4 eigenvalues according to BFB package;
and we know that eigenvalues must be positive, so;
and I can find constraints on parameters.
positivity test In last issue, you mentioned and had some examples about positivity test.
and in the definition, you wrote "Tests a given Higgs potential for postive (semi)definiteness on large field values."
In Positivity Test, I myself must give some numbers to parameters? For example lh = 0.129, ls = 3, lsh = 2, and run it? And for last question, where is the output of Positivity Test? I think it is very obvious and Positivity Test is not necessary, am I wrong? I am a little confused about this!!!
Best Wishes Layla