The 'God particle' referred to in the movie is the Higgs-Boson particle, theorized in 1963-64. The LHC in Geneva is indeed seeking out the nature of this particle, which is claimed to give mass to all matter in the universe.
During filming in Rome, the crew and equipment were blocking the passage of a bridal party on its way to the church for a wedding. Upon discovering this situation, Tom Hanks personally escorted the party through the filming area and rallied the crew to remove the obstacles in the way. After the bride was delivered to the church (on time), her grateful family invited Hanks and director Ron Howard to stay for the wedding ceremony. Unfortunately, Hanks and Howard's busy filming schedule prevented them from accepting the offer.
Crew members visited Vatican City as tourists and extensively photographed the city to capture as much detail as possible, knowing they were unlikely to be allowed to film there, so that they could recreate the sets as faithfully as possible.
Ewan McGregor suggested to Ron Howard that he could do the sequence where the Camerlengo tells the Cardinals about the Illuminati entirely without breaking it into several segments, as he felt that he could pull it off, having known all of his lines. Howard agreed to his suggestion and proceeded to film it. The take was very good, much to the applause of everyone, that it was the only shot taken, and used in the final film.
Ron Howard has said in interviews that he plans on completing a Robert Langdon trilogy, by filming the next story in the series, "The Lost Symbol." For unknown reasons, The Lost Symbol was scrapped during pre-production, and Sony decided to adapt the fourth book, 'Inferno' that had been released in the meanwhile. Inferno (2016) was released in October 2016.
In the previous film, Langdon remarks to Sophie Neuveu that perhaps she cured his claustrophobia by laying her hands upon him. It appears that she did so indeed, as in this movie Langdon is twice confined in a small, glass room with poor ventilation and seems un-fazed. He also seems fine when he needs to squeeze through a small alley between two very close walls in his search for the bomb. However, in the sequel Inferno (2016), there's a scene where Robert mentions that he is claustrophobic.
As well as providing the voice over for the film's teaser trailer (using an American accent), Alfred Molina (who played Aringarosa in the previous movie The Da Vinci Code (2006)) provides the opening voice over for Angels & Demons (2009), this time in his native English accent.
When Langdon is in his office speaking with the Vatican employee, you can see a "Phillips Exeter Academy" flag on the wall. Phillips Exeter is the school where Angels & Demons author Dan Brown taught and once attended, located in New Hampshire, and where Langdon is written to have attended in the novels.
The final piece before the end credits is Dr. Morten Lauridsen's "Lux Aeterna: Introitus" - Dr. Lauridsen has never before allowed this piece to be used in a movie, due to the context being inappropriate to his meaning behind the piece.
This movie became the first big-screen casualty of the Hollywood writer's strike in 2007, because Akiva Goldsman's script still needed work, and he was on strike with the Writers Guild of America. So, production of the movie had to be postponed.
Many of the sets were a combination of physical set pieces and green screens with the backgrounds to be digitally added later. This technique was used for scenes that the crew was unable to film at either due to the volume of tourists, or because the Church denied them permission.
As with the other two films in the trilogy, author Dan Brown named main character Robert Langdon after John Langdon, a close friend and typography master who worked with Brown on ambigram designs for his book, "Angels & Demons", as well as the films. John Langdon also designed an ambigram that was used in the movie Monkeyshine (2008).
The theme "Chevaliers De Sangreal" by Hans Zimmer, in The Da Vinci Code (2006) was used again in the film now called "503" and later it was re-used in Inferno (2016) now called "Life Must Have Its Own Mysteries". Hans Zimmer composer of the three films, called it as the official Robert Langdon theme.
The trivia items below may give away important plot points.
Only film in the trilogy, where Robert is not pursued by the police or a company. In The Da Vinci Code (2006) he is pursued by Captain Fache for being a murder suspect, and in Inferno (2016) he is pursued by the World Health Organization and the Consortium, trying to find a virus that may cause a plague.
In the original novel, it is revealed that the Camerlengo killed the Pope because the latter admitted to having fathered a child. It is subsequently revealed that the child in question, born through artificial insemination, is in fact the Camerlengo. This subplot was omitted from the film, but it's hinted at when the Camerlengo describes the late Pope as a father figure.