star trek - Is there any significance of the colors of Starfleet Uniform?

Starfleet has the three uniforms - blue, gold and red in TNG onwards. I know that these each represent different areas, but is there a canon reason for why these colors were used for each area? e.g. why science is given blue? Please note - I am not asking about the change of uniform, but instead am asking about the canon significance of these colour in the ST universe.

Answer

In the TNG Era, based upon the DS9 Tech Manual

Red for operations in TNG is given as the color of starfleet uniforms in general.

Blue for sciences was retained/returned from prior eras.

Gold¹ was returned because it was the traditional third color, but since Red replaced it for Operations, Technical services got gold instead.

While there's no official canon on it, the licensed games have spoken to it, citing that there were complaints that it was too broad to use just the 3 colors.

We see a large number of different colors in TWOK and later - the left sleeve cuff stripe and the right shoulder strap are in specialty colors.

Roddenberry actually was combining late 1800's USN speciality colors, in simplification, with the carrier practice of colored shirts on the flight deck.

Gold¹ was the traditional color for line officer's stripes, and hence is the color for line officer tunics

Red was used for Naval Engineers. It was also used for other technical specialists.

Blue was used for sciences, mimicking the old USN cobalt blue for Medical and Light Blue for Civil Engineers.

Sparkly Silver/Gray is seen as a flashback for an academy uniform ("Shore Leave" - the cadet is created from Kirk's memories, so this might not be a contemporary uniform to the TOS era).

While not in films as such, Roddenberry spoke of the Line, Restricted Line, and Staff Officer distinctions. Roddenberry also spoke of historical colors of USN uniforms in one interview. Roddenberry was historically minded - he saw Star Trek as a Space Western, and so it wouldn't be too odd for him to reach back to the historical colors of the USN.

As an army officer, Roddenberry was also personally familiar with color distinctions - all US services except the Marine Corps have used color distinctions for specialists at some historical points (for the USAF, it was before the Blue uniforms were adopted). During Roddenberry's time in service, the use of colored ascots with fatigues was common (unit authorized), full dress uniforms used colored backdrops on the officer rank insignia, and colored aiguillettes on the service dress uniform; enlisted had not yet acquired the colored rings behind the US insignia worn on service dress.

We see red service uniforms, all black instructor uniforms, gray and white admiral's full dress uniforms, and colored shirt ship uniforms.

The colors are the same as for TOS.

The second movie has moved to a slate gray dress uniform; it is unclear if this is a service or full dress uniform.

The DVD extras imply that the red service uniform is inspired by the main timeline TOS uniforms. The admirals' Gray & White is a clear homage to ST:TMP.

Line:                     no flashing color.
Medical:                  cobalt blue (1869-1883), maroon (1883-1918)
Pay:                  white (1869-1918)
Engineer:                 red (1869-1899)
Naval Constructor:        dark violet (1872-1918)
Professor of Mathematics: olive green (1872-1918)
Civil Engineer:           light blue (1881-1918)
Dental:                   orange (1913-1918)
Medical Reserve:          crimson (1913-1918)

Chaplain:                 black (June 16 - November 16, 1918)

The flashing color was worn between the officer's rank cuff stripes, or for passed midshipmen and later for ensign, to both sides of the single stripe. Line officers had no such trimming. It was also used on the flashings of the collar; white flashings were worn by line officers.

This practice was discontinued in 1918; at that same point, the USN adopted what would remain the basic pattern of uniforms and insignia through the present. At present, corps colors are used only in unit heraldry. Duty group assignments on carrier aircrews are indicated by colored turtleneck shirts.

¹ Gold: In TOS, it wasn't actually gold. It was green. How he came up with green, however... Truth is, the references to Gold Tunics are from extended universe sources, not from the actual footage. They are golden colored in the Animated series as well. This may, however, be derived from the carrier shirts.