# Stephen Blake

Bristol, United Kingdom

stebla.pwp.blueyonder.co.uk

Age: 59

I work in an engineering company doing algorithms and optics. In my spare time, I'm trying to understand group representation theory.

 May 25 asked What goes wrong when one tries to quantize a scalar field with Fermi statistics? May 11 comment Complex energy eigenstates of the harmonic oscillatorI think your observations on $\langle p|p\rangle=\infty$ are completely correct. I suppose that a complex eigenvalue always means the wavefunction blows up at infinity so the solution is not physically interesting because the particle is at infinity. May 11 revised Complex energy eigenstates of the harmonic oscillatorAdded an explanation of how the energy eigenvalue need not be real in spite of the Hamiltonian being Hermitian. May 11 answered Complex energy eigenstates of the harmonic oscillator May 5 answered Why Quantum Mechanics as a non-fundamental effective theory? Mar 30 answered Heisenberg evolution equation for $\hat{\phi}$ Mar 24 comment Physical and Geometrical interpretation of Differential Forms@Muphrid : I'm not saying a physical quantity has to transform under GL(n,R) to be physical. I'm saying that a physical quantity must transform under one group; the difficulty with $ab=a\bullet b + a\wedge b$ is that each piece on the RHS transforms under a different group. One can see this clearly when GA texts consider projective geometry where all the physical quantities - points, lines, etc. - transform under GL(n,R) so the dot product - transforming under O(n) - cannot appear. One finds that GA texts only use the wedge product in chapters on projective geometry. Mar 24 comment Physical and Geometrical interpretation of Differential Forms@Muphrid : Your linear operator $T$ is a GL(n,R) matrix, and the wedge product term $a \wedge b$ transforms under $T$ as you have written it, but the dot product term $a\bullet b$ does not transform under $T\in GL(n,R)$ so the GA product $ab$ does not make sense as a physical quantity. GA gets around this by only using $ab$ when $T$ is restricted to O(n), but this feels disjointed to me. Your reference to gauge invariance is not relevant to this basic stuff which appears in the first few pages of any GA textbook. Mar 24 comment Physical and Geometrical interpretation of Differential Forms@Muphhrid : A physical quantity has to transform as a representation of a group, but in the GA expression $ab=a\bullet b+a\wedge b$ the dot product transforms under the orthogonal group O(n) whilst the wedge product transforms under the general linear group GL(n,R) : how can mixing groups in this way be a helpful formalism in physics? Mar 11 comment Pure state - density matrix - real life example of boxes in warehouse@nate : You are not crazy, it is the opposite of the quote. The quote was from Lubos' answer and my comment was to challenge its veracity. Mar 11 comment Pure state - density matrix - real life example of boxes in warehouse"Physically, there can't be any objective answer to the question whether a physical system is in a pure state or a mixed state." Suppose I'm given a beam of spin 1/2 particles and a Stern-Gerlach apparatus. I measure the average spins along three axes. The 2x2 density matrix is Hermitian so it has three parameters which are fixed by the measurements. The density matrix is now known and an eigenvalue decomposition says if it's pure or mixed. All inertial observers would agree with the eigenvalues so the result is objective. Mar 4 comment Physical and Geometrical interpretation of Differential FormsDifferential forms are simply antisymmetric tensors; their use makes one forget all the other kinds of physical quantities which cannot be written as antisymmetric tensors. In my opinion, it is better to work with tensors. Feb 28 comment Two ways to form SU(2) singlets?@QuantumDot : I learnt this from Wu-Ki Tung's book "Group Theory in Physics". Feb 28 revised Two ways to form SU(2) singlets?MathJax script slowed down, so I made the post in two goes. Feb 28 answered Two ways to form SU(2) singlets? Feb 14 comment Why and how does symmetry work in circuits?One corner is grounded, and 1V is applied to the node on the opposite corner to the grounded node. I'm not sure if it's appropriate to try to post the spice deck in the comments? Feb 14 comment Why and how does symmetry work in circuits?Are there 40 resistors in your circuit? Feb 14 comment Why and how does symmetry work in circuits?Are you sure the answer is correct? I put the 4x4 mesh into the circuit simulator ngspice and got the resistance as RT=2.136364R whereas your value is RT=1.958333R . Feb 11 awarded Commentator Feb 11 comment Are group representations possible when the solution space is not a vector space?@Edward Hughes - here is an example of a non-linear realization of the symplectic group in the optics of Gaussian beams.